diff --git a/docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases.md b/docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases.md
index c88e70bc6dbe..04facd9a68b9 100644
--- a/docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases.md
+++ b/docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases.md
@@ -65,7 +65,7 @@ julia> default_ordering(R)
 degrevlex([x, y, z])
 
 julia> F = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> default_ordering(F)
 degrevlex([x, y, z])*lex([gen(1), gen(2)])
@@ -140,7 +140,7 @@ julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 2 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> f = (5*x*y^2-y^10+3)*F[1]+(4*x^3+2*y) *F[2]+16*x*F[3]
 (5*x*y^2 - y^10 + 3)*e[1] + (4*x^3 + 2*y)*e[2] + 16*x*e[3]
diff --git a/docs/src/CommutativeAlgebra/GroebnerBases/orderings.md b/docs/src/CommutativeAlgebra/GroebnerBases/orderings.md
index 7fd4a9ec1619..6ebaf9325209 100644
--- a/docs/src/CommutativeAlgebra/GroebnerBases/orderings.md
+++ b/docs/src/CommutativeAlgebra/GroebnerBases/orderings.md
@@ -410,7 +410,7 @@ basis vectors as *lex*, and to the $i > j$ ordering as *invlex*. And, we use the
 julia> R, (w, x, y, z) = polynomial_ring(QQ, ["w", "x", "y", "z"]);
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 4 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> o1 = degrevlex(R)*invlex(gens(F))
 degrevlex([w, x, y, z])*invlex([gen(1), gen(2), gen(3)])
diff --git a/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md b/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md
index 56edbf36b5d1..83822ec9042d 100644
--- a/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md
+++ b/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md
@@ -52,25 +52,21 @@ julia> range(C)
 5:-1:3
 
 julia> C[5]
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> e[1]
-by Submodule with 1 generator
+by submodule with 1 generator
 1 -> x^4*e[1]
 
 julia> delta = map(C, 5)
-Map with following data
-Domain:
-=======
-Subquotient of Submodule with 1 generator
-1 -> e[1]
-by Submodule with 1 generator
-1 -> x^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 1 generator
-1 -> e[1]
-by Submodule with 1 generator
-1 -> x^3*e[1]
+Module homomorphism
+  from subquotient of submodule with 1 generator
+  1 -> e[1]
+  by submodule with 1 generator
+  1 -> x^4*e[1]
+  to subquotient of submodule with 1 generator
+  1 -> e[1]
+  by submodule with 1 generator
+  1 -> x^3*e[1]
 
 julia> matrix(delta)
 [x^2]
diff --git a/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md b/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md
index 6c6496c223fd..ab59e789edb1 100644
--- a/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md
+++ b/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md
@@ -148,7 +148,7 @@ julia> f = x*F[1] + y*F[3]
 x*e[1] + y*e[3]
 
 julia> parent(f)
-Free module of rank 3 over Multivariate polynomial ring in 2 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> coordinates(f)
 Sparse row with positions [1, 3] and values QQMPolyRingElem[x, y]
diff --git a/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md b/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md
index 70790c1a9042..6bf0e31e0790 100644
--- a/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md
+++ b/docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md
@@ -86,7 +86,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x; y]
 [x]
@@ -98,10 +98,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -136,9 +136,9 @@ julia> relations(M)
  z^4*e[1]
 
 julia> ambient_module(M)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -181,7 +181,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x; y]
 [x]
@@ -193,10 +193,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -237,7 +237,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x; y]
 [x]
@@ -249,10 +249,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -261,10 +261,10 @@ julia> m = z*M[1] + M[2]
 (x*z + y)*e[1]
 
 julia> parent(m)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -285,7 +285,7 @@ julia> parent(fm) === ambient_free_module(M)
 true
 
 julia> F = ambient_free_module(M)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> f = x*F[1]
 x*e[1]
diff --git a/experimental/Schemes/CoherentSheaves.jl b/experimental/Schemes/CoherentSheaves.jl
index 715fcb4b19fb..ba23b9b8ac79 100644
--- a/experimental/Schemes/CoherentSheaves.jl
+++ b/experimental/Schemes/CoherentSheaves.jl
@@ -722,10 +722,10 @@ Coherent sheaf of modules
     3: [(s0//s2), (s1//s2), (s3//s2)]   affine 3-space
     4: [(s0//s3), (s1//s3), (s2//s3)]   affine 3-space
 with restrictions
-  1: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
-  2: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
-  3: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
-  4: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+  1: free module of rank 1 over multivariate polynomial ring
+  2: free module of rank 1 over multivariate polynomial ring
+  3: free module of rank 1 over multivariate polynomial ring
+  4: free module of rank 1 over multivariate polynomial ring
 ```
 """
 function twisting_sheaf(IP::AbsProjectiveScheme{<:Field}, d::Int)
@@ -783,10 +783,10 @@ Coherent sheaf of modules
     3: [(s0//s2), (s1//s2), (s3//s2)]   affine 3-space
     4: [(s0//s3), (s1//s3), (s2//s3)]   affine 3-space
 with restrictions
-  1: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
-  2: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
-  3: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
-  4: free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+  1: free module of rank 1 over multivariate polynomial ring
+  2: free module of rank 1 over multivariate polynomial ring
+  3: free module of rank 1 over multivariate polynomial ring
+  4: free module of rank 1 over multivariate polynomial ring
 ```
 """
 function tautological_bundle(IP::AbsProjectiveScheme{<:Field})
diff --git a/src/Modules/FreeModules-graded.jl b/src/Modules/FreeModules-graded.jl
index 864b8ce6ddf4..c07f1395243a 100644
--- a/src/Modules/FreeModules-graded.jl
+++ b/src/Modules/FreeModules-graded.jl
@@ -76,8 +76,9 @@ function show(io::IO, F::FreeModule_dec)
   @show_name(io, F)
   @show_special(io, F)
 
+  io_compact = IOContext(io, :compact => true, :supercompact => true)
   print(io, "Free module of rank $(length(F.d)) over ")
-  print(IOContext(io, :compact =>true), F.R)
+  print(io_compact, F.R)
   if is_graded(F.R)
     print(io, ", graded as ")
   else
@@ -92,7 +93,7 @@ function show(io::IO, F::FreeModule_dec)
     else
       first = false
     end
-    print(IOContext(io, :compact => true), F.R, "^$v(", k, ")")
+    print(io_compact, F.R, "^$v(", k, ")")
   end
 
 #=
@@ -103,8 +104,8 @@ function show(io::IO, F::FreeModule_dec)
     while i+j <= dim(F) && d == F.d[i+j]
       j += 1
     end
-    print(IOContext(io, :compact => true), F.R, "^$j")
-    print(IOContext(io, :compact => true), "(", -d, ")")
+    print(io_compact, F.R, "^$j")
+    print(io_compact, "(", -d, ")")
     if i+j < dim(F)
       print(io, " + ")
     end
diff --git a/src/Modules/ModuleTypes.jl b/src/Modules/ModuleTypes.jl
index 02decdce8796..3431e085039c 100644
--- a/src/Modules/ModuleTypes.jl
+++ b/src/Modules/ModuleTypes.jl
@@ -103,7 +103,7 @@ julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 2 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> f = F(sparse_row(R, [(1,x),(3,y)]))
 x*e[1] + y*e[3]
@@ -272,10 +272,10 @@ julia> B = R[x^2; x*y; y^2; z^4]
 [z^4]
 
 julia> M = SubquoModule(A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> x^2*e[1]
 2 -> x*y*e[1]
 3 -> y^2*e[1]
diff --git a/src/Modules/ModulesGraded.jl b/src/Modules/ModulesGraded.jl
index f40a173310d1..8577725b59dc 100644
--- a/src/Modules/ModulesGraded.jl
+++ b/src/Modules/ModulesGraded.jl
@@ -649,11 +649,13 @@ julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
  z*e[2]
 
 julia> a = hom(F, G, V)
-F -> G
-e[1] -> y*e[1]
-e[2] -> x*e[1] + y*e[2]
-e[3] -> z*e[2]
 Graded module homomorphism of degree [1]
+  from graded free module R^3([0]) of rank 3 over R
+  to graded free module R^2([0]) of rank 2 over R
+defined by
+  e[1] -> y*e[1]
+  e[2] -> x*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> degree(a)
 [1]
@@ -712,11 +714,13 @@ julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
  z*e[2]
 
 julia> a = hom(F, G, V)
-F -> G
-e[1] -> y*e[1]
-e[2] -> x*e[1] + y*e[2]
-e[3] -> z*e[2]
 Graded module homomorphism of degree [1]
+  from graded free module R^3([0]) of rank 3 over R
+  to graded free module R^2([0]) of rank 2 over R
+defined by
+  e[1] -> y*e[1]
+  e[2] -> x*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> is_graded(a)
 true
@@ -748,11 +752,13 @@ julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
  z*e[2]
 
 julia> a = hom(F, G, V)
-F -> G
-e[1] -> y*e[1]
-e[2] -> x*e[1] + y*e[2]
-e[3] -> z*e[2]
 Graded module homomorphism of degree [1]
+  from graded free module R^3([0]) of rank 3 over R
+  to graded free module R^2([0]) of rank 2 over R
+defined by
+  e[1] -> y*e[1]
+  e[2] -> x*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> is_graded(a)
 true
@@ -790,11 +796,13 @@ julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
  z*e[2]
 
 julia> a = hom(F, G, V)
-F -> G
-e[1] -> y*e[1]
-e[2] -> x*e[1] + y*e[2]
-e[3] -> z*e[2]
 Graded module homomorphism of degree [1]
+  from graded free module R^3([0]) of rank 3 over R
+  to graded free module R^2([0]) of rank 2 over R
+defined by
+  e[1] -> y*e[1]
+  e[2] -> x*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> is_homogeneous(a)
 false
@@ -1101,10 +1109,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> degree(a)
 [2]
@@ -1168,10 +1190,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> is_graded(a)
 true
@@ -1203,10 +1239,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> grading_group(a)
 GrpAb: Z
@@ -1241,10 +1291,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> is_homogeneous(a)
 false
@@ -1953,8 +2017,9 @@ function show(io::IO, F::FreeMod_dec)
   @show_name(io, F)
   @show_special(io, F)
 
+  io_compact = IOContext(io, :compact => true, :supercompact => true)
   print(io, "Decorated free module of rank $(rank(F)) over ")
-  print(IOContext(io, :compact =>true), base_ring(F))
+  print(io_compact, base_ring(F))
 
   i = 1
   while i < dim(F)
@@ -1963,8 +2028,8 @@ function show(io::IO, F::FreeMod_dec)
     while i+j <= dim(F) && d == F.d[i+j]
       j += 1
     end
-    print(IOContext(io, :compact => true), base_ring(F), "^$j")
-    print(IOContext(io, :compact => true), "(", -d, ")")
+    print(io_compact, base_ring(F), "^$j")
+    print(io_compact, "(", -d, ")")
     if i+j < dim(F)
       print(io, " + ")
     end
@@ -2697,9 +2762,9 @@ Ideal generated by
   y^3
 
 julia> quotient_ring_as_module(I)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> e[1]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 ```
@@ -2751,7 +2816,7 @@ julia> ideal_as_module(I)
 Submodule with 2 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 ```
 ```jldoctest
 julia> S, (x, y) = graded_polynomial_ring(QQ, ["x", "y"]);
diff --git a/src/Modules/UngradedModules/FreeMod.jl b/src/Modules/UngradedModules/FreeMod.jl
index eaf843d31646..4120676c8eab 100644
--- a/src/Modules/UngradedModules/FreeMod.jl
+++ b/src/Modules/UngradedModules/FreeMod.jl
@@ -37,7 +37,7 @@ The string `name` specifies how the basis vectors are printed.
 julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
 
 julia> FR = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> x*FR[1]
 x*e[1]
@@ -49,7 +49,7 @@ julia> U = complement_of_prime_ideal(P);
 julia> RL, _ = localization(R, U);
 
 julia> FRL = free_module(RL, 2, "f")
-Free module of rank 2 over Localization of multivariate polynomial ring in 3 variables over QQ at complement of prime ideal (x, y, z)
+Free module of rank 2 over localized ring
 
 julia> RL(x)*FRL[1]
 x*f[1]
@@ -65,7 +65,7 @@ x*g[1]
 julia> RQL, _ = localization(RQ, U);
 
 julia> FRQL =  free_module(RQL, 2, "h")
-Free module of rank 2 over Localization of quotient of multivariate polynomial ring at complement of prime ideal
+Free module of rank 2 over localized quotient of multivariate polynomial ring
 
 julia> RQL(x)*FRQL[1]
 x*h[1]
@@ -107,8 +107,9 @@ end
 function show(io::IO, F::FreeMod)
   @show_name(io, F)
   @show_special(io, F)
-  compact = get(io, :compact, false)
-  io_compact = IOContext(io, :compact => true)
+  io = pretty(io)
+  compact = get(io, :supercompact, false)
+  io_compact = IOContext(io, :compact => true, :supercompact => true)
   if is_graded(F)
       if !compact
         print(io, "Graded free module ")
@@ -133,17 +134,17 @@ function show(io::IO, F::FreeMod)
       end
 
       if !compact
-          print(io," of rank $(rank(F)) over ")
+          print(io," of rank $(rank(F)) over ", Lowercase())
           print(io_compact, base_ring(F))
       end
   else
       if !compact
           #Todo: Use once the printing of rings is fixed
           #print(io_compact, "Free module ", base_ring(F), "^$(F.n) of rank $(F.n) over ")
-          print(io_compact, "Free module of rank $(F.n) over ")
+          print(io_compact, "Free module of rank $(rank(F)) over ", Lowercase())
           print(io_compact, F.R)
       else
-          print(io_compact, base_ring(F), "^$(F.n)")
+          print(io_compact, base_ring(F), "^$(rank(F))")
       end
   end
 end
diff --git a/src/Modules/UngradedModules/FreeModElem.jl b/src/Modules/UngradedModules/FreeModElem.jl
index 6c86428ef130..37f1fb08ed6c 100644
--- a/src/Modules/UngradedModules/FreeModElem.jl
+++ b/src/Modules/UngradedModules/FreeModElem.jl
@@ -84,7 +84,7 @@ julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
 julia> F = FreeMod(R,3)
-Free module of rank 3 over Multivariate polynomial ring in 2 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> f = x*gen(F,1)+y*gen(F,3)
 x*e[1] + y*e[3]
diff --git a/src/Modules/UngradedModules/FreeModuleHom.jl b/src/Modules/UngradedModules/FreeModuleHom.jl
index 4e59682070eb..ef3674903da4 100644
--- a/src/Modules/UngradedModules/FreeModuleHom.jl
+++ b/src/Modules/UngradedModules/FreeModuleHom.jl
@@ -42,10 +42,10 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> G = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]];
 
@@ -101,10 +101,10 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> G = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
 3-element Vector{FreeModElem{QQMPolyRingElem}}:
@@ -113,13 +113,9 @@ julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
  z*e[2]
 
 julia> a = hom(F, G, V)
-Map with following data
-Domain:
-=======
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Module homomorphism
+  from free module of rank 3 over multivariate polynomial ring
+  to free module of rank 2 over multivariate polynomial ring
 
 julia> a(F[2])
 x*e[1] + y*e[2]
@@ -130,13 +126,9 @@ julia> B = R[y 0; x y; 0 z]
 [0   z]
 
 julia> b = hom(F, G, B)
-Map with following data
-Domain:
-=======
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Module homomorphism
+  from free module of rank 3 over multivariate polynomial ring
+  to free module of rank 2 over multivariate polynomial ring
 
 julia> a == b
 true
@@ -158,11 +150,13 @@ julia> V1 = [y*G1[1], (x+y)*G1[1]+y*G1[2], z*G1[2]]
  z*e[2]
 
 julia> a1 = hom(F1, G1, V1)
-F1 -> G1
-e[1] -> y*e[1]
-e[2] -> (x + y)*e[1] + y*e[2]
-e[3] -> z*e[2]
 Graded module homomorphism of degree [1]
+  from graded free module Rg^3([0]) of rank 3 over Rg
+  to graded free module Rg^2([0]) of rank 2 over Rg
+defined by
+  e[1] -> y*e[1]
+  e[2] -> (x + y)*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> F2 = graded_free_module(Rg, [1,1,1])
 Graded free module Rg^3([-1]) of rank 3 over Rg
@@ -177,11 +171,13 @@ julia> V2 = [y*G2[1], (x+y)*G2[1]+y*G2[2], z*G2[2]]
  z*e[2]
 
 julia> a2 = hom(F2, G2, V2)
-F2 -> G2
-e[1] -> y*e[1]
-e[2] -> (x + y)*e[1] + y*e[2]
-e[3] -> z*e[2]
 Homogeneous module homomorphism
+  from graded free module Rg^3([-1]) of rank 3 over Rg
+  to graded free module Rg^2([0]) of rank 2 over Rg
+defined by
+  e[1] -> y*e[1]
+  e[2] -> (x + y)*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> B = Rg[y 0; x+y y; 0 z]
 [    y   0]
@@ -189,11 +185,13 @@ julia> B = Rg[y 0; x+y y; 0 z]
 [    0   z]
 
 julia> b = hom(F2, G2, B)
-F2 -> G2
-e[1] -> y*e[1]
-e[2] -> (x + y)*e[1] + y*e[2]
-e[3] -> z*e[2]
 Homogeneous module homomorphism
+  from graded free module Rg^3([-1]) of rank 3 over Rg
+  to graded free module Rg^2([0]) of rank 2 over Rg
+defined by
+  e[1] -> y*e[1]
+  e[2] -> (x + y)*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> a2 == b
 true
@@ -275,41 +273,38 @@ function morphism_type(
   return FreeModuleHom{DomainType, CodomainType, RingMapType}
 end
 
-function Base.show(io::IO, ::MIME"text/plain", fmh::FreeModuleHom{T1, T2, RingMapType}) where {T1 <: AbstractFreeMod, T2 <: ModuleFP, RingMapType}
-  # HACK
-  show(io, fmh)
+function AbstractAlgebra.show_map_data(io::IO, fmh::FreeModuleHom{T1, T2, RingMapType}) where {T1 <: AbstractFreeMod, T2 <: ModuleFP, RingMapType}
+  if is_graded(fmh)
+    println(io)
+    print(io, "defined by", Indent())
+    io_compact = IOContext(io, :compact => true, :supercompact => true)
+    domain_gens = gens(domain(fmh))
+    for g in domain_gens
+      println(io)
+      print(io_compact, g, " -> ", fmh(g))
+    end
+  end
 end
 
 function Base.show(io::IO, fmh::FreeModuleHom{T1, T2, RingMapType}) where {T1 <: AbstractFreeMod, T2 <: ModuleFP, RingMapType}
-  compact = get(io, :compact, false)
-  io_compact = IOContext(io, :compact => true)
-  if is_graded(fmh)  
-    print(io_compact, domain(fmh))
-    print(io, " -> ")
-    print(io_compact, codomain(fmh))
-    if !compact
-      print(io, "\n")
-      for i in 1:ngens(domain(fmh))
-        print(io, domain(fmh)[i], " -> ")
-        print(io_compact, fmh(domain(fmh)[i]))
-        print(io, "\n")
-      end
+  if get(io, :supercompact, false)
+    if is_graded(fmh)
       A = grading_group(fmh)
       if degree(fmh) == A[0]
         print(io, "Homogeneous module homomorphism")
       else
+        io_compact = IOContext(io, :compact => true, :supercompact => true)
         print(io_compact, "Graded module homomorphism of degree ", degree(fmh))
-        print(io, "\n")
       end
+    else
+        print(io, "Module homomorphism")
     end
   else
-    println(io, "Map with following data")
-    println(io, "Domain:")
-    println(io, "=======")
-    println(io, domain(fmh))
-    println(io, "Codomain:")
-    println(io, "=========")
-    print(io, codomain(fmh))
+    io = pretty(io)
+    print(io, "Hom: ")
+    io_compact = IOContext(io, :compact => true, :supercompact => true)
+    print(io_compact, Lowercase(), domain(fmh), " -> ")
+    print(io_compact, Lowercase(), codomain(fmh))
   end
 end
 
@@ -324,22 +319,18 @@ that converts elements from $S$ into morphisms $F \to G$.
 julia> R, _ = polynomial_ring(QQ, ["x", "y", "z"]);
 
 julia> F1 = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> F2 = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> V, f = hom(F1, F2)
-(hom of (F1, F2), Map: hom of (F1, F2) -> set of all homomorphisms from Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ to Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ)
+(hom of (F1, F2), Map: hom of (F1, F2) -> set of all homomorphisms from Free module of rank 3 over multivariate polynomial ring to Free module of rank 2 over multivariate polynomial ring)
 
 julia> f(V[1])
-Map with following data
-Domain:
-=======
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Module homomorphism
+  from free module of rank 3 over multivariate polynomial ring
+  to free module of rank 2 over multivariate polynomial ring
 
 ```
 
@@ -356,11 +347,13 @@ julia> V, f = hom(F1, F2)
 (hom of (F1, F2), Map: hom of (F1, F2) -> set of all homomorphisms from Graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg to Graded free module Rg^1([-3]) + Rg^1([-5]) of rank 2 over Rg)
 
 julia> f(V[1])
-F1 -> F2
-e[1] -> e[1]
-e[2] -> 0
-e[3] -> 0
 Graded module homomorphism of degree [2]
+  from graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg
+  to graded free module Rg^1([-3]) + Rg^1([-5]) of rank 2 over Rg
+defined by
+  e[1] -> e[1]
+  e[2] -> 0
+  e[3] -> 0
 
 ```
 """
@@ -413,10 +406,10 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> G = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]];
 
@@ -425,15 +418,9 @@ julia> a = hom(F, G, V);
 julia> kernel(a)
 (Submodule with 1 generator
 1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
-represented as subquotient with no relations., Map with following data
-Domain:
-=======
-Submodule with 1 generator
+represented as subquotient with no relations, Hom:
 1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
-represented as subquotient with no relations.
-Codomain:
-=========
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ)
+represented as subquotient with no relations -> F)
 ```
 
 ```jldoctest
@@ -450,11 +437,9 @@ julia> a = hom(F, G, V);
 julia> kernel(a)
 (Graded submodule of F
 1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
-represented as subquotient with no relations, Graded submodule of F
+represented as subquotient with no relations, Hom: graded submodule of F
 1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
-represented as subquotient with no relations -> F
-x*z*e[1] - y*z*e[2] + y^2*e[3] -> x*z*e[1] - y*z*e[2] + y^2*e[3]
-Homogeneous module homomorphism)
+represented as subquotient with no relations -> F)
 
 ```
 """
@@ -503,10 +488,10 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> G = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]];
 
@@ -517,17 +502,11 @@ julia> image(a)
 1 -> y*e[1]
 2 -> x*e[1] + y*e[2]
 3 -> z*e[2]
-represented as subquotient with no relations., Map with following data
-Domain:
-=======
-Submodule with 3 generators
+represented as subquotient with no relations, Hom:
 1 -> y*e[1]
 2 -> x*e[1] + y*e[2]
 3 -> z*e[2]
-represented as subquotient with no relations.
-Codomain:
-=========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ)
+represented as subquotient with no relations -> G)
 ```
 
 ```jldoctest
@@ -546,15 +525,11 @@ julia> image(a)
 1 -> y*e[1]
 2 -> x*e[1] + y*e[2]
 3 -> z*e[2]
-represented as subquotient with no relations, Graded submodule of G
+represented as subquotient with no relations, Hom: graded submodule of G
 1 -> y*e[1]
 2 -> x*e[1] + y*e[2]
 3 -> z*e[2]
-represented as subquotient with no relations -> G
-y*e[1] -> y*e[1]
-x*e[1] + y*e[2] -> x*e[1] + y*e[2]
-z*e[2] -> z*e[2]
-Homogeneous module homomorphism)
+represented as subquotient with no relations -> G)
 
 ```
 """
diff --git a/src/Modules/UngradedModules/FreeResolutions.jl b/src/Modules/UngradedModules/FreeResolutions.jl
index dee8ee23e56a..61fc3642b723 100644
--- a/src/Modules/UngradedModules/FreeResolutions.jl
+++ b/src/Modules/UngradedModules/FreeResolutions.jl
@@ -36,7 +36,7 @@ function free_show(io::IO, C::ComplexOfMorphisms)
     push!(rank_mod, rank(M))
   end
 
-  io = IOContext(io, :compact => true)
+  io = IOContext(io, :compact => true, :supercompact => true)
   N = get_attribute(C, :free_res)
   if N !== nothing
     print(io, "Free resolution")
@@ -106,10 +106,10 @@ julia> B = R[x^2; x*y; y^2; z^4]
 [z^4]
 
 julia> M = SubquoModule(A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> x^2*e[1]
 2 -> x*y*e[1]
 3 -> y^2*e[1]
@@ -276,10 +276,10 @@ julia> B = R[x^2; x*y; y^2; z^4]
 [z^4]
 
 julia> M = SubquoModule(A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> x^2*e[1]
 2 -> x*y*e[1]
 3 -> y^2*e[1]
@@ -294,7 +294,7 @@ julia> is_complete(fr)
 false
 
 julia> fr[4]
-Free module of rank 0 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 0 over multivariate polynomial ring
 
 julia> fr
 Free resolution of M
diff --git a/src/Modules/UngradedModules/Hom_and_ext.jl b/src/Modules/UngradedModules/Hom_and_ext.jl
index 8bd774eff373..0276ba7bdfb8 100644
--- a/src/Modules/UngradedModules/Hom_and_ext.jl
+++ b/src/Modules/UngradedModules/Hom_and_ext.jl
@@ -23,10 +23,10 @@ julia> F = FreeMod(R, 2);
 julia> V = [x*F[1], y^2*F[2]];
 
 julia> M = quo(F, V)[1]
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x*e[1]
 2 -> y^2*e[2]
 
@@ -116,10 +116,10 @@ julia> F = FreeMod(R, 2);
 julia> V = [x*F[1], y^2*F[2]];
 
 julia> M = quo(F, V)[1]
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x*e[1]
 2 -> y^2*e[2]
 
@@ -138,23 +138,19 @@ julia> relations(H)
  y^2*(e[2] -> e[2])
 
 julia> a = element_to_homomorphism(H[1]+y*H[2])
-Map with following data
-Domain:
-=======
-Subquotient of Submodule with 2 generators
-1 -> e[1]
-2 -> e[2]
-by Submodule with 2 generators
-1 -> x*e[1]
-2 -> y^2*e[2]
-Codomain:
-=========
-Subquotient of Submodule with 2 generators
-1 -> e[1]
-2 -> e[2]
-by Submodule with 2 generators
-1 -> x*e[1]
-2 -> y^2*e[2]
+Module homomorphism
+  from subquotient of submodule with 2 generators
+  1 -> e[1]
+  2 -> e[2]
+  by submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y^2*e[2]
+  to subquotient of submodule with 2 generators
+  1 -> e[1]
+  2 -> e[2]
+  by submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y^2*e[2]
 
 julia> matrix(a)
 [1   0]
@@ -182,10 +178,10 @@ julia> F = FreeMod(R, 2);
 julia> V = [x*F[1], y^2*F[2]];
 
 julia> M = quo(F, V)[1]
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x*e[1]
 2 -> y^2*e[2]
 
diff --git a/src/Modules/UngradedModules/HomologicalAlgebra.jl b/src/Modules/UngradedModules/HomologicalAlgebra.jl
index fcf37755b1ad..c74ee7b4b293 100644
--- a/src/Modules/UngradedModules/HomologicalAlgebra.jl
+++ b/src/Modules/UngradedModules/HomologicalAlgebra.jl
@@ -204,26 +204,26 @@ julia> F = free_module(R, 1);
 julia> Q, _ = quo(F, [x*F[1]]);
 
 julia> T0 = tor(Q, M, 0)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1] \otimes e[1]
 2 -> y*e[1] \otimes e[1]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> x^2*e[1] \otimes e[1]
 2 -> y^3*e[1] \otimes e[1]
 3 -> z^4*e[1] \otimes e[1]
 4 -> x*y*e[1] \otimes e[1]
 
 julia> T1 = tor(Q, M, 1)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> -x*e[1] \otimes e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1] \otimes e[1]
 2 -> y^3*e[1] \otimes e[1]
 3 -> z^4*e[1] \otimes e[1]
 
 julia> T2 =  tor(Q, M, 2)
 Submodule with 0 generators
-represented as subquotient with no relations.
+represented as subquotient with no relations
 ```
 """
 function tor(M::ModuleFP, N::ModuleFP, i::Int)
@@ -459,18 +459,18 @@ julia> C = ComplexOfMorphisms(ModuleFP, [a, b]);
 
 julia> H = homology(C)
 3-element Vector{SubquoModule{QQMPolyRingElem}}:
- Subquotient of Submodule with 1 generator
+ Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 1 generator
+by submodule with 1 generator
 1 -> x^4*e[1]
- Subquotient of Submodule with 1 generator
+ Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x^3*e[1]
 2 -> x^2*e[1]
- Subquotient of Submodule with 1 generator
+ Subquotient of submodule with 1 generator
 1 -> e[1]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x^3*e[1]
 2 -> x^2*e[1]
 ```
@@ -506,9 +506,9 @@ julia> b = hom(B, B, [x^2*B[1]]);
 julia> C = ComplexOfMorphisms(ModuleFP, [a, b]);
 
 julia> H = homology(C, 1)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x^3*e[1]
 2 -> x^2*e[1]
 ```
@@ -550,40 +550,40 @@ julia> F = FreeMod(R, 1);
 julia> V = [x*F[1], y*F[1]];
 
 julia> M = quo(F, V)[1]
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> e[1]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
 
 julia> ext(M, M, 0)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> (e[1] -> e[1])
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x*(e[1] -> e[1])
 2 -> y*(e[1] -> e[1])
 
 julia> ext(M, M, 1)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> (e[2] -> e[1])
 2 -> (e[1] -> e[1])
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> x*(e[1] -> e[1])
 2 -> y*(e[1] -> e[1])
 3 -> x*(e[2] -> e[1])
 4 -> y*(e[2] -> e[1])
 
 julia> ext(M, M, 2)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> (e[1] -> e[1])
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x*(e[1] -> e[1])
 2 -> y*(e[1] -> e[1])
 3 -> -x*(e[1] -> e[1])
 
 julia> ext(M, M, 3)
 Submodule with 0 generators
-represented as subquotient with no relations.
+represented as subquotient with no relations
 ```
 """
 function ext(M::ModuleFP, N::ModuleFP, i::Int)
diff --git a/src/Modules/UngradedModules/Methods.jl b/src/Modules/UngradedModules/Methods.jl
index 48bbdd6ee98a..f9fc5baf7bc9 100644
--- a/src/Modules/UngradedModules/Methods.jl
+++ b/src/Modules/UngradedModules/Methods.jl
@@ -19,10 +19,24 @@ julia> B = Rg[x^2; y^3; z^4];
 julia> M = SubquoModule(F, A, B);
 
 julia> is_equal_with_morphism(M, M)
-M -> M
-x*e[1] -> x*e[1]
-y*e[1] -> y*e[1]
 Homogeneous module homomorphism
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*e[1]
+  y*e[1] -> y*e[1]
 ```
 """
 function is_equal_with_morphism(M::SubquoModule{T}, N::SubquoModule{T}, task::Symbol = :none) where {T}
diff --git a/src/Modules/UngradedModules/Presentation.jl b/src/Modules/UngradedModules/Presentation.jl
index 98e2dcf66d6d..f105ade504fa 100644
--- a/src/Modules/UngradedModules/Presentation.jl
+++ b/src/Modules/UngradedModules/Presentation.jl
@@ -141,22 +141,28 @@ julia> p[1]
 Graded free module Rg^0 of rank 0 over Rg
 
 julia> map(p,-1)
-F -> 0
-e[1] -> 0
-e[2] -> 0
-e[3] -> 0
 Homogeneous module homomorphism
+  from graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg
+  to graded free module Rg^0 of rank 0 over Rg
+defined by
+  e[1] -> 0
+  e[2] -> 0
+  e[3] -> 0
 
 julia> map(p,0)
-F -> F
-e[1] -> e[1]
-e[2] -> e[2]
-e[3] -> e[3]
 Homogeneous module homomorphism
+  from graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg
+  to graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg
+defined by
+  e[1] -> e[1]
+  e[2] -> e[2]
+  e[3] -> e[3]
 
 julia> map(p,1)
-0 -> F
 Homogeneous module homomorphism
+  from graded free module Rg^0 of rank 0 over Rg
+  to graded free module Rg^1([-1]) + Rg^2([-2]) of rank 3 over Rg
+defined by
 
 julia> F = graded_free_module(Rg, 1);
 
@@ -188,25 +194,43 @@ julia> P[1]
 Graded free module Rg^2([-2]) + Rg^1([-3]) + Rg^2([-5]) of rank 5 over Rg
 
 julia> map(P,-1)
-M -> 0
-x*e[1] -> 0
-y*e[1] -> 0
 Homogeneous module homomorphism
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded free module Rg^0 of rank 0 over Rg
+defined by
+  x*e[1] -> 0
+  y*e[1] -> 0
 
 julia> map(P,0)
-Rg^2 -> M
-e[1] -> x*e[1]
-e[2] -> y*e[1]
 Homogeneous module homomorphism
+  from graded free module Rg^2([-1]) of rank 2 over Rg
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  e[1] -> x*e[1]
+  e[2] -> y*e[1]
 
 julia> map(P,1)
-Rg^5 -> Rg^2
-e[1] -> x*e[1]
-e[2] -> -y*e[1] + x*e[2]
-e[3] -> y^2*e[2]
-e[4] -> z^4*e[1]
-e[5] -> z^4*e[2]
 Homogeneous module homomorphism
+  from graded free module Rg^2([-2]) + Rg^1([-3]) + Rg^2([-5]) of rank 5 over Rg
+  to graded free module Rg^2([-1]) of rank 2 over Rg
+defined by
+  e[1] -> x*e[1]
+  e[2] -> -y*e[1] + x*e[2]
+  e[3] -> y^2*e[2]
+  e[4] -> z^4*e[1]
+  e[5] -> z^4*e[2]
 ```
 """
 function presentation(M::ModuleFP)
@@ -235,19 +259,19 @@ julia> A = R[x; y];
 julia> B = R[x^2; y^3; z^4];
 
 julia> M = SubquoModule(A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
 
 julia> C = present_as_cokernel(M)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 5 generators
+by submodule with 5 generators
 1 -> x*e[1]
 2 -> -y*e[1] + x*e[2]
 3 -> y^2*e[2]
@@ -275,7 +299,7 @@ by submodule of Rg^2 generated by
 2 -> -y*e[1] + x*e[2]
 3 -> y^2*e[2]
 4 -> z^4*e[1]
-5 -> z^4*e[2], Graded subquotient of submodule of Rg^2 generated by
+5 -> z^4*e[2], Hom: graded subquotient of submodule of Rg^2 generated by
 1 -> e[1]
 2 -> e[2]
 by submodule of Rg^2 generated by
@@ -283,10 +307,7 @@ by submodule of Rg^2 generated by
 2 -> -y*e[1] + x*e[2]
 3 -> y^2*e[2]
 4 -> z^4*e[1]
-5 -> z^4*e[2] -> M
-e[1] -> x*e[1]
-e[2] -> y*e[1]
-Homogeneous module homomorphism)
+5 -> z^4*e[2] -> M)
 ```
 """
 function present_as_cokernel(SQ::SubquoModule, task::Symbol = :none)
@@ -332,25 +353,21 @@ If `task = :only_morphism`, return only an isomorphism.
 julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
 
 julia> F = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> present_as_cokernel(F)
 Submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 
 julia> present_as_cokernel(F, :only_morphism)
-Map with following data
-Domain:
-=======
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Submodule with 2 generators
-1 -> e[1]
-2 -> e[2]
-represented as subquotient with no relations.
+Module homomorphism
+  from free module of rank 2 over multivariate polynomial ring
+  to submodule with 2 generators
+  1 -> e[1]
+  2 -> e[2]
+  represented as subquotient with no relations
 ```
 """
 function present_as_cokernel(F::FreeMod, task::Symbol = :none)
@@ -381,28 +398,16 @@ that `N` is generated by a minimal number of elements.
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);
 
 julia> M = SubquoModule(identity_matrix(R, 2), R[1 x+y])
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 1 generator
+by submodule with 1 generator
 1 -> e[1] + (x + y)*e[2]
 
 julia> N, phi = prune_with_map(M)
 (Submodule with 1 generator
 1 -> e[1]
-represented as subquotient with no relations., Map with following data
-Domain:
-=======
-Submodule with 1 generator
-1 -> e[1]
-represented as subquotient with no relations.
-Codomain:
-=========
-Subquotient of Submodule with 2 generators
-1 -> e[1]
-2 -> e[2]
-by Submodule with 1 generator
-1 -> e[1] + (x + y)*e[2])
+represented as subquotient with no relations, Hom: N -> M)
 
 julia> phi(first(gens(N)))
 e[2]
diff --git a/src/Modules/UngradedModules/SubModuleOfFreeModule.jl b/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
index e243d3695794..f38f8dde6e64 100644
--- a/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
+++ b/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
@@ -295,8 +295,8 @@ end
 function show(io::IO, M::SubModuleOfFreeModule)
   @show_name(io, M)
   @show_special(io, M)
-  io_compact = IOContext(io, :compact => true)
-  compact = get(io, :compact, false)
+  io_compact = IOContext(io, :compact => true, :supercompact => true)
+  compact = get(io, :supercompact, false)
   if !compact
     if is_graded(M)
       print(io_compact, "Graded submodule of ", M.F)
diff --git a/src/Modules/UngradedModules/SubQuoHom.jl b/src/Modules/UngradedModules/SubQuoHom.jl
index 0b2394f43eda..dbb8070b8a7d 100644
--- a/src/Modules/UngradedModules/SubQuoHom.jl
+++ b/src/Modules/UngradedModules/SubQuoHom.jl
@@ -41,42 +41,38 @@ function SubQuoHom(D::SubquoModule, C::ModuleFP{T}, mat::MatElem{T}, h::RingMapT
   end
 end
 
-function Base.show(io::IO, ::MIME"text/plain", fmh::SubQuoHom{T1, T2, RingMapType}) where {T1 <: AbstractSubQuo, T2 <: ModuleFP, RingMapType}
-  # HACK
-  show(io, fmh)
+function AbstractAlgebra.show_map_data(io::IO, fmh::SubQuoHom{T1, T2, RingMapType}) where {T1 <: AbstractSubQuo, T2 <: ModuleFP, RingMapType}
+  if is_graded(fmh)
+    println(io)
+    print(io, "defined by", Indent())
+    io_compact = IOContext(io, :compact => true, :supercompact => true)
+    domain_gens = gens(domain(fmh))
+    for g in domain_gens
+      println(io)
+      print(io_compact, g, " -> ", fmh(g))
+    end
+  end
 end
 
 function Base.show(io::IO, fmh::SubQuoHom{T1, T2, RingMapType}) where {T1 <: AbstractSubQuo, T2 <: ModuleFP, RingMapType}
-  compact = get(io, :compact, false)
-  io_compact = IOContext(io, :compact => true)
-  domain_gens = gens(domain(fmh))
-  if is_graded(fmh)
-    print(io_compact, domain(fmh))
-    print(io, " -> ")
-    print(io_compact, codomain(fmh))
-    if !compact
-      print(io, "\n")
-      for i in 1:length(domain_gens)
-        print(io, domain_gens[i], " -> ")
-        print(io_compact, fmh(domain_gens[i]))
-        print(io, "\n")
-      end
+  if get(io, :supercompact, false)
+    if is_graded(fmh)
       A = grading_group(fmh)
       if degree(fmh) == A[0]
         print(io, "Homogeneous module homomorphism")
       else
+        io_compact = IOContext(io, :compact => true, :supercompact => true)
         print(io_compact, "Graded module homomorphism of degree ", degree(fmh))
-        print(io, "\n")
       end
+    else
+        print(io, "Module homomorphism")
     end
   else
-    println(io, "Map with following data")
-    println(io, "Domain:")
-    println(io, "=======")
-    println(io, domain(fmh))
-    println(io, "Codomain:")
-    println(io, "=========")
-    print(io, codomain(fmh))
+    io = pretty(io)
+    print(io, "Hom: ")
+    io_compact = IOContext(io, :compact => true, :supercompact => true)
+    print(io_compact, Lowercase(), domain(fmh), " -> ")
+    print(io_compact, Lowercase(), codomain(fmh))
   end
 end
 
@@ -121,7 +117,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x; y]
 [x]
@@ -133,10 +129,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -149,25 +145,21 @@ julia> V = [y^2*N[1], x*N[2]]
  x*y*e[1]
 
 julia> a = hom(M, N, V)
-Map with following data
-Domain:
-=======
-Subquotient of Submodule with 2 generators
-1 -> x*e[1]
-2 -> y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 2 generators
-1 -> x*e[1]
-2 -> y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
+Module homomorphism
+  from subquotient of submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to subquotient of submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
 
 julia> is_welldefined(a)
 true
@@ -187,7 +179,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x; y];
 
@@ -231,10 +223,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> is_welldefined(a)
 true
@@ -244,10 +250,24 @@ julia> W = Rg[y^2 0; 0 x^2]
 [  0   x^2]
 
 julia> b = hom(M, N, W)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> a == b
 true
@@ -258,10 +278,24 @@ julia> W = [y*N[1], x*N[2]]
  x*y*e[1]
 
 julia> c = hom(M, N, W)
-M -> M
-x*e[1] -> x*y*e[1]
-y*e[1] -> x*y*e[1]
 Graded module homomorphism of degree [1]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y*e[1]
+  y*e[1] -> x*y*e[1]
 
 julia> is_welldefined(c)
 false
@@ -369,7 +403,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x; y]
 [x]
@@ -381,10 +415,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -426,10 +460,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> matrix(a)
 [y^2     0]
@@ -575,20 +623,16 @@ Submodule with 3 generators
 1 -> y*e[1]
 2 -> x*e[1] + y*e[2]
 3 -> z*e[2]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 
 julia> incl
-Map with following data
-Domain:
-=======
-Submodule with 3 generators
-1 -> y*e[1]
-2 -> x*e[1] + y*e[2]
-3 -> z*e[2]
-represented as subquotient with no relations.
-Codomain:
-=========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Module homomorphism
+  from submodule with 3 generators
+  1 -> y*e[1]
+  2 -> x*e[1] + y*e[2]
+  3 -> z*e[2]
+  represented as subquotient with no relations
+  to free module of rank 2 over multivariate polynomial ring
 ```
 
 ```jldoctest
@@ -606,10 +650,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -626,34 +670,30 @@ julia> a = hom(M, N, V);
 julia> I, incl = image(a);
 
 julia> I
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*y^2*e[1]
 2 -> x*y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
 
 julia> incl
-Map with following data
-Domain:
-=======
-Subquotient of Submodule with 2 generators
-1 -> x*y^2*e[1]
-2 -> x*y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 2 generators
-1 -> x*e[1]
-2 -> y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
+Module homomorphism
+  from subquotient of submodule with 2 generators
+  1 -> x*y^2*e[1]
+  2 -> x*y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to subquotient of submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
 ```
 
 ```jldoctest
@@ -679,10 +719,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> image(a)
 (Graded subquotient of submodule of F generated by
@@ -691,16 +745,13 @@ julia> image(a)
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1], Graded subquotient of submodule of F generated by
+3 -> z^4*e[1], Hom: graded subquotient of submodule of F generated by
 1 -> x*y^2*e[1]
 2 -> x^2*y*e[1]
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1] -> M
-x*y^2*e[1] -> x*y^2*e[1]
-x^2*y*e[1] -> x^2*y*e[1]
-Homogeneous module homomorphism)
+3 -> z^4*e[1] -> M)
 ```
 """
 function image(a::ModuleFPHom)
@@ -755,18 +806,14 @@ julia> K, incl = kernel(a);
 julia> K
 Submodule with 1 generator
 1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 
 julia> incl
-Map with following data
-Domain:
-=======
-Submodule with 1 generator
-1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
-represented as subquotient with no relations.
-Codomain:
-=========
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Module homomorphism
+  from submodule with 1 generator
+  1 -> x*z*e[1] - y*z*e[2] + y^2*e[3]
+  represented as subquotient with no relations
+  to free module of rank 3 over multivariate polynomial ring
 ```
 
 ```jldoctest
@@ -784,10 +831,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -804,36 +851,32 @@ julia> a = hom(M, N, V);
 julia> K, incl = kernel(a);
 
 julia> K
-Subquotient of Submodule with 3 generators
+Subquotient of submodule with 3 generators
 1 -> (-x + y^2)*e[1]
 2 -> x*y*e[1]
 3 -> -x*y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
 
 julia> incl
-Map with following data
-Domain:
-=======
-Subquotient of Submodule with 3 generators
-1 -> (-x + y^2)*e[1]
-2 -> x*y*e[1]
-3 -> -x*y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 2 generators
-1 -> x*e[1]
-2 -> y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
+Module homomorphism
+  from subquotient of submodule with 3 generators
+  1 -> (-x + y^2)*e[1]
+  2 -> x*y*e[1]
+  3 -> -x*y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to subquotient of submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
 ```
 
 ```jldoctest
@@ -859,10 +902,24 @@ julia> N = M;
 julia> V = [y^2*N[1], x^2*N[2]];
 
 julia> a = hom(M, N, V)
-M -> M
-x*e[1] -> x*y^2*e[1]
-y*e[1] -> x^2*y*e[1]
 Graded module homomorphism of degree [2]
+  from graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to graded subquotient of submodule of F generated by
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule of F generated by
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+defined by
+  x*e[1] -> x*y^2*e[1]
+  y*e[1] -> x^2*y*e[1]
 
 julia> kernel(a)
 (Graded subquotient of submodule of F generated by
@@ -872,18 +929,14 @@ julia> kernel(a)
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1], Graded subquotient of submodule of F generated by
+3 -> z^4*e[1], Hom: graded subquotient of submodule of F generated by
 1 -> -y*e[1]
 2 -> (x^2 - y^2)*e[1]
 3 -> -x*y*e[1]
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1] -> M
--y*e[1] -> -y*e[1]
-(x^2 - y^2)*e[1] -> (x^2 - y^2)*e[1]
--x*y*e[1] -> -x*y*e[1]
-Homogeneous module homomorphism)
+3 -> z^4*e[1] -> M)
 
 ```
 """
diff --git a/src/Modules/UngradedModules/SubquoModule.jl b/src/Modules/UngradedModules/SubquoModule.jl
index d0b43401c21b..5fa17e9bbd52 100644
--- a/src/Modules/UngradedModules/SubquoModule.jl
+++ b/src/Modules/UngradedModules/SubquoModule.jl
@@ -42,7 +42,7 @@ julia> R, (x,y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
 julia> F = FreeMod(R,2)
-Free module of rank 2 over Multivariate polynomial ring in 2 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> O = [x*F[1]+F[2],y*F[2]]
 2-element Vector{FreeModElem{QQMPolyRingElem}}:
@@ -53,7 +53,7 @@ julia> M = SubquoModule(F, O)
 Submodule with 2 generators
 1 -> x*e[1] + e[2]
 2 -> y*e[2]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 
 ```
 """
@@ -134,10 +134,10 @@ julia> B = R[x^2; x*y; y^2; z^4]
 [z^4]
 
 julia> M = SubquoModule(A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> x^2*e[1]
 2 -> x*y*e[1]
 3 -> y^2*e[1]
@@ -187,7 +187,7 @@ free module homomorphisms with codomain `F` represented by `A` and `B`.
 julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
 
 julia> FR = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> AR = R[x; y]
 [x]
@@ -199,10 +199,10 @@ julia> BR = R[x^2; y^3; z^4]
 [z^4]
 
 julia> MR = SubquoModule(FR, AR, BR)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -214,7 +214,7 @@ julia> U = complement_of_prime_ideal(P);
 julia> RL, _ = localization(R, U);
 
 julia> FRL = free_module(RL, 1)
-Free module of rank 1 over Localization of multivariate polynomial ring in 3 variables over QQ at complement of prime ideal (x, y, z)
+Free module of rank 1 over localized ring
 
 julia> ARL = RL[x; y]
 [x]
@@ -226,10 +226,10 @@ julia> BRL = RL[x^2; y^3; z^4]
 [z^4]
 
 julia> MRL = SubquoModule(FRL, ARL, BRL)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -249,10 +249,10 @@ julia> BRQ = RQ[x^2; y^3; z^4]
 [z^4]
 
 julia> MRQ = SubquoModule(FRQ, ARQ, BRQ)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> 2*x^2*e[1]
 3 -> z^4*e[1]
@@ -260,7 +260,7 @@ by Submodule with 3 generators
 julia> RQL, _ = localization(RQ, U);
 
 julia> FRQL = free_module(RQL, 1)
-Free module of rank 1 over Localization of quotient of multivariate polynomial ring at complement of prime ideal
+Free module of rank 1 over localized quotient of multivariate polynomial ring
 
 julia> ARQL = RQL[x; y]
 [x]
@@ -272,10 +272,10 @@ julia> BRQL = RQL[x^2; y^3; z^4]
 [z^4]
 
 julia> MRQL = SubquoModule(FRQL, ARQL, BRQL)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> 0
 2 -> 0
 3 -> z^4*e[1]
@@ -301,16 +301,20 @@ julia> V2 = [z*G[2]+y*G[1]]
  y*e[1] + z*e[2]
 
 julia> a1 = hom(F1, G, V1)
-F1 -> G
-e[1] -> y*e[1]
-e[2] -> (x + y)*e[1] + y*e[2]
-e[3] -> z*e[2]
 Homogeneous module homomorphism
+  from graded free module Rg^3([-2]) of rank 3 over Rg
+  to graded free module Rg^2([-1]) of rank 2 over Rg
+defined by
+  e[1] -> y*e[1]
+  e[2] -> (x + y)*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> a2 = hom(F2, G, V2)
-F2 -> G
-e[1] -> y*e[1] + z*e[2]
 Homogeneous module homomorphism
+  from graded free module Rg^1([-2]) of rank 1 over Rg
+  to graded free module Rg^2([-1]) of rank 2 over Rg
+defined by
+  e[1] -> y*e[1] + z*e[2]
 
 julia> V = subquotient(a1,a2)
 Graded subquotient of submodule of G generated by
@@ -356,15 +360,19 @@ subquotient(A::MatElem{T}, B::MatElem{T}) where {T} = SubquoModule(A, B)
 function show(io::IO, SQ::SubquoModule)
   @show_name(io, SQ)
   @show_special(io, SQ)
-  io_compact = IOContext(io, :compact => true)
+  io = pretty(io)
+  io_compact = IOContext(io, :compact => true, :supercompact => true)
 
   if is_graded(SQ)
       if isdefined(SQ, :quo) && !iszero(SQ.quo)
           print(io, "Graded subquotient")
-          print(io_compact, " of submodule of ", SQ.F, " generated by", SQ.sub, "\nby submodule of ", SQ.F, " generated by", SQ.quo)
+          println(io_compact, " of submodule of ", Lowercase(), SQ.F,
+                              " generated by", Lowercase(), SQ.sub)
+          print(io_compact, "by submodule of ", Lowercase(), SQ.F,
+                            " generated by", Lowercase(), SQ.quo)
       else
-          print(io_compact, "Graded submodule of ", SQ.F)
-          print(io_compact, SQ.sub, "\n")
+          print(io_compact, "Graded submodule of ", Lowercase(), SQ.F)
+          println(io_compact, Lowercase(), SQ.sub)
           print(io, "represented as subquotient with no relations")
       end
   else
@@ -378,10 +386,11 @@ function show(io::IO, SQ::SubquoModule)
       #     print(io, "represented as subquotient with no relations")
       # end
       if isdefined(SQ, :quo) && !iszero(SQ.quo)
-        print(io, "Subquotient of ", SQ.sub, "\nby ", SQ.quo)
+        println(io, "Subquotient of ", Lowercase(), SQ.sub, )
+        print(io, "by ", Lowercase(), SQ.quo)
       else
-        print(io, SQ.sub, "\n")
-        print(io, "represented as subquotient with no relations.")
+        println(io, SQ.sub)
+        print(io, "represented as subquotient with no relations")
       end
   end
 end
@@ -394,7 +403,7 @@ Show `SQ` as a subquotient of *matrices* `A` and `B`.
 function show_subquo(SQ::SubquoModule)
   #@show_name(io, SQ)
   #@show_special(io, SQ)
-  io_compact = IOContext(stdout, :compact => true)
+  io_compact = IOContext(stdout, :supercompact => true)
   if isdefined(SQ, :quo)
     if is_generated_by_standard_unit_vectors(SQ.sub)
       if is_graded(SQ)
@@ -425,7 +434,7 @@ function show_subquo(SQ::SubquoModule)
 end
 
 function show_morphism_as_map(f::ModuleFPHom, print_non_zero_only = false)
-  io_compact = IOContext(stdout, :compact => true)
+  io_compact = IOContext(stdout, :supercompact => true)
   print(io_compact, domain(f), " -> ", codomain(f))
   print("\n")
   D = domain(f)
@@ -474,10 +483,10 @@ julia> W = R[y 0; x y; 0 z]
 julia> a = hom(F, G, W);
 
 julia> cokernel(a)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> y*e[1]
 2 -> x*e[1] + y*e[2]
 3 -> z*e[2]
@@ -498,10 +507,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -516,10 +525,10 @@ julia> V = [y^2*N[1], x*N[2]]
 julia> a = hom(M, N, V);
 
 julia> cokernel(a)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 5 generators
+by submodule with 5 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -540,11 +549,13 @@ julia> W = Rg[y 0; x y; 0 z]
 [0   z]
 
 julia> a = hom(F, G, W)
-F -> G
-e[1] -> y*e[1]
-e[2] -> x*e[1] + y*e[2]
-e[3] -> z*e[2]
 Graded module homomorphism of degree [1]
+  from graded free module Rg^3([0]) of rank 3 over Rg
+  to graded free module Rg^2([0]) of rank 2 over Rg
+defined by
+  e[1] -> y*e[1]
+  e[2] -> x*e[1] + y*e[2]
+  e[3] -> z*e[2]
 
 julia> M = cokernel(a)
 Graded subquotient of submodule of G generated by
@@ -571,17 +582,17 @@ Return the cokernel of `A` as an object of type `SubquoModule` with ambient free
 julia> R, (x,y,z) = polynomial_ring(QQ, ["x", "y", "z"]);
 
 julia> F = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> A = R[x y; 2*x^2 3*y^2]
 [    x       y]
 [2*x^2   3*y^2]
  
 julia> M = cokernel(F, A)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x*e[1] + y*e[2]
 2 -> 2*x^2*e[1] + 3*y^2*e[2]
 
@@ -635,10 +646,10 @@ julia> A = R[x y; 2*x^2 3*y^2]
 [2*x^2   3*y^2]
  
 julia> M = cokernel(A)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 2 generators
+by submodule with 2 generators
 1 -> x*e[1] + y*e[2]
 2 -> 2*x^2*e[1] + 3*y^2*e[2]
 
@@ -658,7 +669,7 @@ Return the image of `A` as an object of type `SubquoModule` with ambient free mo
 julia> R, (x,y,z) = polynomial_ring(QQ, ["x", "y", "z"]);
 
 julia> F = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> A = R[x y; 2*x^2 3*y^2]
 [    x       y]
@@ -668,7 +679,7 @@ julia> M = image(F, A)
 Submodule with 2 generators
 1 -> x*e[1] + y*e[2]
 2 -> 2*x^2*e[1] + 3*y^2*e[2]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 
 julia> ambient_free_module(M) === F
 true
@@ -720,7 +731,7 @@ julia> M = image(A)
 Submodule with 2 generators
 1 -> x*e[1] + y*e[2]
 2 -> 2*x^2*e[1] + 3*y^2*e[2]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 ```
 """
 function image(A::MatElem)
@@ -838,7 +849,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> AM = R[x;]
 [x]
@@ -849,9 +860,9 @@ julia> BM = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, AM, BM)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -866,10 +877,10 @@ julia> BN = R[x^2+y^4; y^3; z^4]
 [      z^4]
 
 julia> N = SubquoModule(F, AN, BN)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> (x^2 + y^4)*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -966,7 +977,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> AM = R[x;]
 [x]
@@ -977,9 +988,9 @@ julia> BM = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, AM, BM)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -994,10 +1005,10 @@ julia> BN = R[x^2+y^4; y^3; z^4]
 [      z^4]
 
 julia> N = SubquoModule(F, AN, BN)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> (x^2 + y^4)*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -1129,10 +1140,7 @@ Graded submodule of F2
 represented as subquotient with no relations
 
 julia> is_canonically_isomorphic_with_map(M1, M2)
-(true, M1 -> M2
-x^3*e[1] + x^2*e[2] + x*e[3] -> x^3*e[1] + x*e[2] + x^2*e[3]
-(2*x^4 + x^3*y)*e[1] + (x^3 + 2*x*y^2)*e[2] + x^2*e[3] -> (2*x^4 + x^3*y)*e[1] + x^2*e[2] + (x^3 + 2*x*y^2)*e[3]
-Homogeneous module homomorphism)
+(true, Hom: M1 -> M2)
 
 ```
 """
@@ -1165,7 +1173,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> AM = R[x;]
 [x]
@@ -1176,9 +1184,9 @@ julia> BM = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, AM, BM)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -1192,9 +1200,9 @@ julia> BN = R[x^2; y^3; z^4]
 [z^4]
 
 julia> N = SubquoModule(F, AN, BN)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -1202,53 +1210,45 @@ by Submodule with 3 generators
 julia> O = sum(M, N);
 
 julia> O[1]
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
 
 julia> O[2]
-Map with following data
-Domain:
-=======
-Subquotient of Submodule with 1 generator
-1 -> x*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 2 generators
-1 -> x*e[1]
-2 -> y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
+Module homomorphism
+  from subquotient of submodule with 1 generator
+  1 -> x*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to subquotient of submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
 
 julia> O[3]
-Map with following data
-Domain:
-=======
-Subquotient of Submodule with 1 generator
-1 -> y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 2 generators
-1 -> x*e[1]
-2 -> y*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
+Module homomorphism
+  from subquotient of submodule with 1 generator
+  1 -> y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  to subquotient of submodule with 2 generators
+  1 -> x*e[1]
+  2 -> y*e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
 ```
 
 ```jldoctest
@@ -1287,23 +1287,19 @@ julia> sum(M, N)
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1], M -> Graded subquotient of submodule of F generated by
+3 -> z^4*e[1], Hom: M -> graded subquotient of submodule of F generated by
 1 -> x*e[1]
 2 -> y*e[1]
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1]
-x*e[1] -> x*e[1]
-Homogeneous module homomorphism, N -> Graded subquotient of submodule of F generated by
+3 -> z^4*e[1], Hom: N -> graded subquotient of submodule of F generated by
 1 -> x*e[1]
 2 -> y*e[1]
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1]
-y*e[1] -> y*e[1]
-Homogeneous module homomorphism)
+3 -> z^4*e[1])
 
 ```
 """
@@ -1347,7 +1343,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> AM = R[x;]
 [x]
@@ -1358,9 +1354,9 @@ julia> BM = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, AM, BM)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -1374,18 +1370,18 @@ julia> BN = R[x^2; y^3; z^4]
 [z^4]
 
 julia> N = SubquoModule(F, AN, BN)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
 
 julia> O = M + N
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -1450,7 +1446,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> AM = R[x;]
 [x]
@@ -1461,9 +1457,9 @@ julia> BM = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, AM, BM)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> x*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -1477,55 +1473,33 @@ julia> BN = R[x^2; y^3; z^4]
 [z^4]
 
 julia> N = SubquoModule(F, AN, BN)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
 
 julia> intersect(M, N)
-(Subquotient of Submodule with 2 generators
+(Subquotient of submodule with 2 generators
 1 -> -x*y*e[1]
 2 -> x*z^4*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1], Map with following data
-Domain:
-=======
-Subquotient of Submodule with 2 generators
+3 -> z^4*e[1], Hom: subquotient of
 1 -> -x*y*e[1]
 2 -> x*z^4*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 1 generator
-1 -> x*e[1]
-by Submodule with 3 generators
+by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1], Map with following data
-Domain:
-=======
-Subquotient of Submodule with 2 generators
+3 -> z^4*e[1] -> M, Hom: subquotient of
 1 -> -x*y*e[1]
 2 -> x*z^4*e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-Codomain:
-=========
-Subquotient of Submodule with 1 generator
-1 -> y*e[1]
-by Submodule with 3 generators
+by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1])
+3 -> z^4*e[1] -> N)
 ```
 
 ```jldoctest
@@ -1564,25 +1538,19 @@ julia> intersect(M, N)
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1], Graded subquotient of submodule of F generated by
+3 -> z^4*e[1], Hom: graded subquotient of submodule of F generated by
 1 -> -x*y*e[1]
 2 -> x*z^4*e[1]
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1] -> M
--x*y*e[1] -> -x*y*e[1]
-x*z^4*e[1] -> x*z^4*e[1]
-Homogeneous module homomorphism, Graded subquotient of submodule of F generated by
+3 -> z^4*e[1] -> M, Hom: graded subquotient of submodule of F generated by
 1 -> -x*y*e[1]
 2 -> x*z^4*e[1]
 by submodule of F generated by
 1 -> x^2*e[1]
 2 -> y^3*e[1]
-3 -> z^4*e[1] -> N
--x*y*e[1] -> x*y*e[1]
-x*z^4*e[1] -> 0
-Homogeneous module homomorphism)
+3 -> z^4*e[1] -> N)
 
 ```
 """
diff --git a/src/Modules/UngradedModules/SubquoModuleElem.jl b/src/Modules/UngradedModules/SubquoModuleElem.jl
index 0de3819bd670..9e341e9d0062 100644
--- a/src/Modules/UngradedModules/SubquoModuleElem.jl
+++ b/src/Modules/UngradedModules/SubquoModuleElem.jl
@@ -543,20 +543,16 @@ Submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
-represented as subquotient with no relations.
+represented as subquotient with no relations
 
 julia> incl
-Map with following data
-Domain:
-=======
-Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
-represented as subquotient with no relations.
-Codomain:
-=========
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Module homomorphism
+  from submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
+  represented as subquotient with no relations
+  to free module of rank 1 over multivariate polynomial ring
 ```
 """
 function sub(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}, task::Symbol = :with_morphism) where T
@@ -711,26 +707,22 @@ julia> V = [x^2*F[1]; y^3*F[1]; z^4*F[1]];
 julia> N, proj = quo(F, V);
 
 julia> N
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
 
 julia> proj
-Map with following data
-Domain:
-=======
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
-Codomain:
-=========
-Subquotient of Submodule with 1 generator
-1 -> e[1]
-by Submodule with 3 generators
-1 -> x^2*e[1]
-2 -> y^3*e[1]
-3 -> z^4*e[1]
+Module homomorphism
+  from free module of rank 1 over multivariate polynomial ring
+  to subquotient of submodule with 1 generator
+  1 -> e[1]
+  by submodule with 3 generators
+  1 -> x^2*e[1]
+  2 -> y^3*e[1]
+  3 -> z^4*e[1]
 ```
 """
 function quo(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}, task::Symbol = :with_morphism) where T
@@ -873,7 +865,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x^2+y^2;]
 [x^2 + y^2]
@@ -884,9 +876,9 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> (x^2 + y^2)*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
@@ -939,7 +931,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over multivariate polynomial ring
 
 julia> A = R[x; y]
 [x]
@@ -951,10 +943,10 @@ julia> B = R[x^2; y^3; z^4]
 [z^4]
 
 julia> M = SubquoModule(F, A, B)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> x*e[1]
 2 -> y*e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> x^2*e[1]
 2 -> y^3*e[1]
 3 -> z^4*e[1]
diff --git a/src/Modules/homological-algebra.jl b/src/Modules/homological-algebra.jl
index 9e11e5215e34..13f85639dbc5 100644
--- a/src/Modules/homological-algebra.jl
+++ b/src/Modules/homological-algebra.jl
@@ -31,10 +31,10 @@ julia> U = matrix([x^3-y^2 o; o x^3-y^2; -x^2 y; -y x])
 [       -y           x]
 
 julia> M = quo(F,U)[1]
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> (x^3 - y^2)*e[1]
 2 -> (x^3 - y^2)*e[2]
 3 -> -x^2*e[1] + y*e[2]
@@ -106,10 +106,10 @@ julia> U = matrix([x^3-y^2 o; o x^3-y^2; -x^2 y; -y x])
 [       -y           x]
 
 julia> M = quo(F,U)[1]
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> (x^3 - y^2)*e[1]
 2 -> (x^3 - y^2)*e[2]
 3 -> -x^2*e[1] + y*e[2]
@@ -163,10 +163,10 @@ julia> U = matrix([x^3-y^2 o; o x^3-y^2; -x^2 y; -y x])
 [       -y           x]
 
 julia> M = quo(F,U)[1]
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> e[1]
 2 -> e[2]
-by Submodule with 4 generators
+by submodule with 4 generators
 1 -> (x^3 - y^2)*e[1]
 2 -> (x^3 - y^2)*e[2]
 3 -> -x^2*e[1] + y*e[2]
@@ -295,26 +295,26 @@ julia> V =  [x*y, x*z, y*z]
  y*z
 
 julia> koszul_homology(V, F, 0)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> e[1]∧e[2]∧e[3]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> y*z*e[1]∧e[2]∧e[3]
 2 -> -x*z*e[1]∧e[2]∧e[3]
 3 -> x*y*e[1]∧e[2]∧e[3]
 
 julia> koszul_homology(V, F, 1)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> y*e[1]∧e[3] \otimes e[1] + z*e[2]∧e[3] \otimes e[1]
 2 -> x*e[1]∧e[2] \otimes e[1] - z*e[2]∧e[3] \otimes e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> -x*z*e[1]∧e[2] \otimes e[1] - y*z*e[1]∧e[3] \otimes e[1]
 2 -> x*y*e[1]∧e[2] \otimes e[1] - y*z*e[2]∧e[3] \otimes e[1]
 3 -> x*y*e[1]∧e[3] \otimes e[1] + x*z*e[2]∧e[3] \otimes e[1]
 
 julia> koszul_homology(V, F, 2)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> x*y*e[1] \otimes e[1] + x*z*e[2] \otimes e[1] + y*z*e[3] \otimes e[1]
-by Submodule with 1 generator
+by submodule with 1 generator
 1 -> x*y*e[1] \otimes e[1] + x*z*e[2] \otimes e[1] + y*z*e[3] \otimes e[1]
 ```
 
@@ -326,26 +326,26 @@ julia> TC = ideal(R, [x*z-y^2, w*z-x*y, w*y-x^2]);
 julia> F = free_module(R, 1);
 
 julia> koszul_homology(gens(TC), F, 0)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> e[1]∧e[2]∧e[3]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> (w*y - x^2)*e[1]∧e[2]∧e[3]
 2 -> (-w*z + x*y)*e[1]∧e[2]∧e[3]
 3 -> (x*z - y^2)*e[1]∧e[2]∧e[3]
 
 julia> koszul_homology(gens(TC), F, 1)
-Subquotient of Submodule with 2 generators
+Subquotient of submodule with 2 generators
 1 -> z*e[1]∧e[2] \otimes e[1] + y*e[1]∧e[3] \otimes e[1] + x*e[2]∧e[3] \otimes e[1]
 2 -> y*e[1]∧e[2] \otimes e[1] + x*e[1]∧e[3] \otimes e[1] + w*e[2]∧e[3] \otimes e[1]
-by Submodule with 3 generators
+by submodule with 3 generators
 1 -> (-w*z + x*y)*e[1]∧e[2] \otimes e[1] + (-w*y + x^2)*e[1]∧e[3] \otimes e[1]
 2 -> (x*z - y^2)*e[1]∧e[2] \otimes e[1] + (-w*y + x^2)*e[2]∧e[3] \otimes e[1]
 3 -> (x*z - y^2)*e[1]∧e[3] \otimes e[1] + (w*z - x*y)*e[2]∧e[3] \otimes e[1]
 
 julia> koszul_homology(gens(TC), F, 2)
-Subquotient of Submodule with 1 generator
+Subquotient of submodule with 1 generator
 1 -> (-x*z + y^2)*e[1] \otimes e[1] + (-w*z + x*y)*e[2] \otimes e[1] + (-w*y + x^2)*e[3] \otimes e[1]
-by Submodule with 1 generator
+by submodule with 1 generator
 1 -> (x*z - y^2)*e[1] \otimes e[1] + (w*z - x*y)*e[2] \otimes e[1] + (w*y - x^2)*e[3] \otimes e[1]
 ```
 """
diff --git a/src/Rings/orderings.jl b/src/Rings/orderings.jl
index 2a627f5e0208..f5a72a323d4a 100644
--- a/src/Rings/orderings.jl
+++ b/src/Rings/orderings.jl
@@ -1541,7 +1541,7 @@ julia> cmp(lex([x,y,z]), z, one(R))
 1
 
 julia> F = free_module(R, 2)
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over multivariate polynomial ring
 
 julia> cmp(lex(R)*invlex(F), F[1], F[2])
 -1
@@ -1826,7 +1826,7 @@ Return the ring ordering induced by `ord`.
 julia> R, (w, x, y, z) = polynomial_ring(QQ, ["w", "x", "y", "z"]);
 
 julia> F = free_module(R, 3)
-Free module of rank 3 over Multivariate polynomial ring in 4 variables over QQ
+Free module of rank 3 over multivariate polynomial ring
 
 julia> o = invlex(gens(F))*degrevlex(R)
 invlex([gen(1), gen(2), gen(3)])*degrevlex([w, x, y, z])
diff --git a/test/Modules/ExteriorPowers.jl b/test/Modules/ExteriorPowers.jl
index 8030b98c2004..9f0f18cf9ff8 100644
--- a/test/Modules/ExteriorPowers.jl
+++ b/test/Modules/ExteriorPowers.jl
@@ -130,7 +130,7 @@ end
   u = (F[1], F[4], F[3])
 
   @test "$(mm(v))" == "e[1]∧e[3]∧e[4]"
-  @test "$(F3)" == "⋀^3(Free module of rank 5 over Multivariate polynomial ring in 5 variables over QQ)"
+  @test "$(F3)" == "⋀^3(Free module of rank 5 over multivariate polynomial ring)"
 
   eu = sum(f*e for (f, e) in zip(gens(R), gens(F)))
   K = koszul_complex(eu)