diff --git a/Project.toml b/Project.toml
index e65f9aec36c1..745b6a756afa 100644
--- a/Project.toml
+++ b/Project.toml
@@ -25,22 +25,22 @@ UUIDs = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"
 cohomCalg_jll = "5558cf25-a90e-53b0-b813-cadaa3ae7ade"
 
 [compat]
-AbstractAlgebra = "0.40.8"
+AbstractAlgebra = "0.41.3"
 AlgebraicSolving = "0.4.11"
 Distributed = "1.6"
 DocStringExtensions = "0.8, 0.9"
 GAP = "0.10.2"
-Hecke = "0.30.11"
+Hecke = "0.31.3"
 JSON = "^0.20, ^0.21"
 JSON3 = "1.13.2"
 LazyArtifacts = "1.6"
-Nemo = "0.43.3"
+Nemo = "0.44.0"
 Pkg = "1.6"
 Polymake = "0.11.14"
 Random = "1.6"
 RandomExtensions = "0.4.3"
 Serialization = "1.6"
-Singular = "0.22.4"
+Singular = "0.23.0"
 TOPCOM_jll = "0.17.8"
 UUIDs = "1.6"
 cohomCalg_jll = "0.32.0"
diff --git a/docs/src/CommutativeAlgebra/rings.md b/docs/src/CommutativeAlgebra/rings.md
index aa09ec1d811b..d9ea584c43b9 100644
--- a/docs/src/CommutativeAlgebra/rings.md
+++ b/docs/src/CommutativeAlgebra/rings.md
@@ -405,7 +405,7 @@ julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
 julia> B = MPolyBuildCtx(R)
-Builder for an element of Multivariate polynomial ring in 2 variables over QQ
+Builder for an element of multivariate polynomial ring
 
 julia> for i = 1:5 push_term!(B, QQ(i), [i, i-1]) end
 
diff --git a/docs/src/NumberTheory/galois.md b/docs/src/NumberTheory/galois.md
index 9294ab41c1b8..0d878cbbd3f8 100644
--- a/docs/src/NumberTheory/galois.md
+++ b/docs/src/NumberTheory/galois.md
@@ -96,10 +96,10 @@ julia> F, a = function_field(x^6 + 108*t^2 + 108*t + 27);
 
 julia> subfields(F)
 4-element Vector{Any}:
- (Function Field over Rational field with defining polynomial a^3 + 54*t + 27, (1//12*_a^4 + (3//2*t + 3//4)*_a)//(t + 1//2))
- (Function Field over Rational field with defining polynomial a^2 + 108*t^2 + 108*t + 27, _a^3)
- (Function Field over Rational field with defining polynomial a^3 - 108*t^2 - 108*t - 27, -_a^2)
- (Function Field over Rational field with defining polynomial a^3 - 54*t - 27, (-1//12*_a^4 + (3//2*t + 3//4)*_a)//(t + 1//2))
+ (Function Field over QQ with defining polynomial a^3 + 54*t + 27, (1//12*_a^4 + (3//2*t + 3//4)*_a)//(t + 1//2))
+ (Function Field over QQ with defining polynomial a^2 + 108*t^2 + 108*t + 27, _a^3)
+ (Function Field over QQ with defining polynomial a^3 - 108*t^2 - 108*t - 27, -_a^2)
+ (Function Field over QQ with defining polynomial a^3 - 54*t - 27, (-1//12*_a^4 + (3//2*t + 3//4)*_a)//(t + 1//2))
 
 julia> galois_group(F)
 (Permutation group of degree 6 and order 6, Galois context for s^6 + 108*t^2 + 540*t + 675)
diff --git a/experimental/GModule/Cohomology.jl b/experimental/GModule/Cohomology.jl
index 2448b469e5a1..77e5bf91e9a3 100644
--- a/experimental/GModule/Cohomology.jl
+++ b/experimental/GModule/Cohomology.jl
@@ -380,8 +380,8 @@ function Oscar.direct_product(C::GModule...; task::Symbol = :none)
   @assert all(x->x.G == G, C)
   mM, pro, inj = direct_product([x.M for x = C]..., task = :both)
 
-  mC = gmodule(G, [direct_sum(mM, mM, [action(C[i], g) for i=1:length(C)]) for g = gens(G)])
-  mC.iac = [direct_sum(mM, mM, [action(C[i], inv(g)) for i=1:length(C)]) for g = gens(G)]
+  mC = gmodule(G, [hom_direct_sum(mM, mM, [action(C[i], g) for i=1:length(C)]) for g = gens(G)])
+  mC.iac = [hom_direct_sum(mM, mM, [action(C[i], inv(g)) for i=1:length(C)]) for g = gens(G)]
 
   if task == :none
     return mC
@@ -403,7 +403,7 @@ function Oscar.tensor_product(C::GModule{<:Any, FinGenAbGroup}...; task::Symbol
   @assert all(x->x.G == C[1].G, C)
 
   T, mT = Oscar.tensor_product([x.M for x = C]...; task = :map)
-  TT = gmodule(T, C[1].G, [hom(T, T, [action(C[i], g) for i=1:length(C)]) for g = gens(C[1].G)])
+  TT = gmodule(T, C[1].G, [hom_tensor(T, T, [action(C[i], g) for i=1:length(C)]) for g = gens(C[1].G)])
   if task == :map
     return TT, mT
   else
@@ -492,6 +492,7 @@ _rank(M) = rank(M)
 
 Oscar.dim(C::GModule) = _rank(C.M)
 Oscar.base_ring(C::GModule) = base_ring(C.M)
+Oscar.base_ring_type(::Type{GModule{gT, mT}}) where {gT, mT} = base_ring_type(mT)
 Oscar.group(C::GModule) = C.G
 
 ###########################################################
diff --git a/experimental/GModule/GModule.jl b/experimental/GModule/GModule.jl
index 01b5c6d4cb4f..34450024d1ff 100644
--- a/experimental/GModule/GModule.jl
+++ b/experimental/GModule/GModule.jl
@@ -40,10 +40,10 @@ julia> C = gmodule(CyclotomicField, C);
 julia> h = subfields(base_ring(C), degree = 2)[1][2];
 
 julia> restriction_of_scalars(C, h)
-G-module for G acting on vector space of dimension 4 over number field of degree 2 over QQ
+G-module for G acting on vector space of dimension 4 over number field
 
 julia> restriction_of_scalars(C, QQ)
-G-module for G acting on vector space of dimension 8 over rational field
+G-module for G acting on vector space of dimension 8 over QQ
 
 ```
 """
diff --git a/experimental/GModule/Misc.jl b/experimental/GModule/Misc.jl
index 54e16c35118f..9bc37f9b1bb3 100644
--- a/experimental/GModule/Misc.jl
+++ b/experimental/GModule/Misc.jl
@@ -2,7 +2,6 @@ module Misc
 using Oscar
 import Base: ==, parent
 
-export coimage
 export relative_field
 
 Hecke.minpoly(a::QQBarFieldElem) = minpoly(Hecke.Globals.Qx, a)
@@ -194,11 +193,6 @@ end
 ## functions that will eventually get defined in Hecke.jl,
 ## and then should get removed here
 
-function Hecke.roots(a::FinFieldElem, i::Int)
-  kx, x = polynomial_ring(parent(a), cached = false)
-  return roots(x^i-a)
-end
-
 function Oscar.dual(h::Map{FinGenAbGroup, FinGenAbGroup})
   A = domain(h)
   B = codomain(h)
@@ -253,27 +247,6 @@ Hecke.extend(::Hecke.QQEmb, mp::MapFromFunc{QQField, AbsSimpleNumField}) = compl
 
 Hecke.restrict(::Hecke.NumFieldEmb, ::Map{QQField, AbsSimpleNumField}) = complex_embeddings(QQ)[1]
 
-"""
-    direct_sum(G::FinGenAbGroup, H::FinGenAbGroup, V::Vector{<:Map{FinGenAbGroup, FinGenAbGroup}})
-
-For groups `G = prod G_i` and `H = prod H_i` as well as maps `V_i: G_i -> H_i`,
-build the induced map from `G -> H`.
-"""
-function Oscar.direct_sum(G::FinGenAbGroup, H::FinGenAbGroup, V::Vector{<:Map{FinGenAbGroup, FinGenAbGroup}})
-  dG = get_attribute(G, :direct_product)
-  dH = get_attribute(H, :direct_product)
-
-  if dG === nothing || dH === nothing
-    error("both groups need to be direct products")
-  end
-  @assert length(V) == length(dG) == length(dH)
-
-  @assert all(i -> domain(V[i]) == dG[i] && codomain(V[i]) == dH[i], 1:length(V))
-  h = hom(G, H, cat([matrix(V[i]) for i=1:length(V)]..., dims=(1,2)), check = !true)
-  return h
-
-end
-
 #XXX: have a type for an implicit field - in Hecke?
 #     add all(?) the other functions to it
 function relative_field(m::Map{<:AbstractAlgebra.Field, <:AbstractAlgebra.Field})
@@ -387,133 +360,17 @@ function Hecke.induce_rational_reconstruction(a::ZZMatrix, pg::ZZRingElem; Error
 end
 
 
-#############################################################################
-##
-## functions that will eventually get defined in Nemo.jl,
-## and then should get removed here
-
-function (k::Nemo.fpField)(a::Vector)
-  @assert length(a) == 1
-  return k(a[1])
-end
-
-function (k::fqPolyRepField)(a::Vector)
-  return k(polynomial(Native.GF(Int(characteristic(k))), a))
-end
-
-
 #############################################################################
 ##
 ## functions that will eventually get defined in AbstractAlgebra.jl,
 ## and then should get removed here
 
-Base.pairs(M::MatElem) = Base.pairs(IndexCartesian(), M)
-Base.pairs(::IndexCartesian, M::MatElem) = Base.Iterators.Pairs(M, CartesianIndices(axes(M)))
-
-Oscar.matrix(phi::Generic.IdentityMap{<:AbstractAlgebra.FPModule}) = identity_matrix(base_ring(domain(phi)), dim(domain(phi)))
-
-Oscar.gen(M::AbstractAlgebra.FPModule, i::Int) = M[i]
-
-Oscar.is_free(M::Generic.FreeModule) = true
 Oscar.is_free(M::Generic.DirectSumModule) = all(is_free, M.m)
 
-function Base.iterate(M::AbstractAlgebra.FPModule{T}) where T <: FinFieldElem
-  k = base_ring(M)
-  if dim(M) == 0
-    return zero(M), iterate([1])
-  end
-  p = Base.Iterators.ProductIterator(Tuple([k for i=1:dim(M)]))
-  f = iterate(p)
-  return M(elem_type(k)[f[1][i] for i=1:dim(M)]), (f[2], p)
-end
-
-function Base.iterate(::AbstractAlgebra.FPModule{<:FinFieldElem}, ::Tuple{Int64, Int64})
-  return nothing
-end
-
-Oscar.issubset(M::AbstractAlgebra.FPModule{T}, N::AbstractAlgebra.FPModule{T}) where T<:RingElement = is_submodule(M, N)
-
-function is_sub_with_data(M::AbstractAlgebra.FPModule{T}, N::AbstractAlgebra.FPModule{T}) where T<:RingElement
-  fl = is_submodule(N, M)
-  if fl
-    return fl, hom(M, N, elem_type(N)[N(m) for m = gens(M)])
-  else
-    return fl, hom(M, N, elem_type(N)[zero(N) for m = gens(M)])
-  end
-end
-
-function Oscar.hom(V::AbstractAlgebra.Module, W::AbstractAlgebra.Module, v::Vector{<:ModuleElem}; check::Bool = true)
-  if ngens(V) == 0
-    return Generic.ModuleHomomorphism(V, W, zero_matrix(base_ring(V), ngens(V), ngens(W)))
-  end
-  return Generic.ModuleHomomorphism(V, W, reduce(vcat, [x.v for x = v]))
-end
-function Oscar.hom(V::AbstractAlgebra.Module, W::AbstractAlgebra.Module, v::MatElem; check::Bool = true)
-  return Generic.ModuleHomomorphism(V, W, v)
-end
-function Oscar.inv(M::Generic.ModuleHomomorphism)
-  return hom(codomain(M), domain(M), inv(matrix(M)))
-end
-
-Oscar.is_finite(M::AbstractAlgebra.FPModule{<:FinFieldElem}) = true
-
-function Oscar.order(F::AbstractAlgebra.FPModule{<:FinFieldElem})
-  return order(base_ring(F))^dim(F)
-end
-
-function Base.iterate(M::AbstractAlgebra.FPModule{T}, st::Tuple{<:Tuple, <:Base.Iterators.ProductIterator}) where T <: FinFieldElem
-  n = iterate(st[2], st[1])
-  if n === nothing
-    return n
-  end
-  return M(elem_type(base_ring(M))[n[1][i] for i=1:dim(M)]), (n[2], st[2])
-end
-
-function Base.length(M::AbstractAlgebra.FPModule{T}) where T <: FinFieldElem
-  return Int(order(M))
-end
-
-function Base.eltype(M::AbstractAlgebra.FPModule{T}) where T <: FinFieldElem
-  return elem_type(M)
-end
-
-function Oscar.dim(M::AbstractAlgebra.Generic.DirectSumModule{<:FieldElem})
-  return sum(dim(x) for x = M.m)
-end
-
-Base.:*(a::T, b::Generic.ModuleHomomorphism{T}) where {T} = hom(domain(b), codomain(b), a * matrix(b))
-Base.:*(a::T, b::Generic.ModuleIsomorphism{T}) where {T} = hom(domain(b), codomain(b), a * matrix(b))
-Base.:+(a::Generic.ModuleHomomorphism, b::Generic.ModuleHomomorphism) = hom(domain(a), codomain(a), matrix(a) + matrix(b))
-Base.:-(a::Generic.ModuleHomomorphism, b::Generic.ModuleHomomorphism) = hom(domain(a), codomain(a), matrix(a) - matrix(b))
-Base.:-(a::Generic.ModuleHomomorphism) = hom(domain(a), codomain(a), -matrix(a))
-
-function Base.:(==)(a::Union{Generic.ModuleHomomorphism, Generic.ModuleIsomorphism}, b::Union{Generic.ModuleHomomorphism, Generic.ModuleIsomorphism})
-  domain(a) === domain(b) || return false
-  codomain(a) === codomain(b) || return false
-  return matrix(a) == matrix(b)
-end
-
-function Base.hash(a::Union{Generic.ModuleHomomorphism, Generic.ModuleIsomorphism}, h::UInt)
-  h = hash(domain(a), h)
-  h = hash(codomain(a), h)
-  h = hash(matrix(a), h)
-  return h
-end
-
 function Oscar.pseudo_inv(h::Generic.ModuleHomomorphism)
   return MapFromFunc(codomain(h), domain(h), x->preimage(h, x))
 end
 
-function Oscar.direct_sum(M::AbstractAlgebra.Generic.DirectSumModule{T}, N::AbstractAlgebra.Generic.DirectSumModule{T}, mp::Vector{AbstractAlgebra.Generic.ModuleHomomorphism{T}})  where T
-  @assert length(M.m) == length(mp) == length(N.m)
-  return hom(M, N, cat(map(matrix, mp)..., dims = (1,2)))
-end
-
-function coimage(h::Map)
-  return quo(domain(h), kernel(h)[1])
-end
-
 end # module
 using .Misc
-export coimage
 export relative_field
diff --git a/experimental/LieAlgebras/src/LieAlgebras.jl b/experimental/LieAlgebras/src/LieAlgebras.jl
index 649dce18f6b4..db8d0b8b4b55 100644
--- a/experimental/LieAlgebras/src/LieAlgebras.jl
+++ b/experimental/LieAlgebras/src/LieAlgebras.jl
@@ -45,8 +45,9 @@ import ..Oscar:
   gen,
   gens,
   height,
-  hom_tensor,
   hom,
+  hom_direct_sum,
+  hom_tensor,
   ideal,
   identity_map,
   image,
@@ -131,7 +132,6 @@ export exterior_power
 export fundamental_weight
 export fundamental_weights
 export general_linear_lie_algebra
-export hom_direct_sum
 export induced_map_on_symmetric_power
 export induced_map_on_tensor_power
 export is_cartan_matrix
@@ -260,7 +260,6 @@ export exterior_power
 export fundamental_weight
 export fundamental_weights
 export general_linear_lie_algebra
-export hom_direct_sum
 export induced_map_on_symmetric_power
 export induced_map_on_tensor_power
 export is_cartan_matrix
diff --git a/experimental/LinearQuotients/src/cox_rings.jl b/experimental/LinearQuotients/src/cox_rings.jl
index 7648d81cc871..d98c0bd66fd8 100644
--- a/experimental/LinearQuotients/src/cox_rings.jl
+++ b/experimental/LinearQuotients/src/cox_rings.jl
@@ -6,6 +6,10 @@
 
 base_ring(HBB::HomBasisBuilder) = HBB.R
 
+base_ring_type(
+  ::Type{HomBasisBuilder{RingType,RingElemType}}
+) where {RingType,RingElemType} = RingType
+
 power_product_cache(HBB::HomBasisBuilder) = HBB.C
 
 group(HBB::HomBasisBuilder) = HBB.G
diff --git a/experimental/LinearQuotients/src/linear_quotients.jl b/experimental/LinearQuotients/src/linear_quotients.jl
index b1b6b88772f6..7b706c3cec1f 100644
--- a/experimental/LinearQuotients/src/linear_quotients.jl
+++ b/experimental/LinearQuotients/src/linear_quotients.jl
@@ -22,6 +22,7 @@ end
 
 group(L::LinearQuotient) = L.group
 base_ring(L::LinearQuotient) = base_ring(group(L))
+base_ring_type(::Type{LinearQuotient{S,T}}) where {S,T} = base_ring_type(MatrixGroup{S,T})
 
 function fixed_root_of_unity(L::LinearQuotient)
   if isdefined(L, :root_of_unity)
diff --git a/experimental/ModStd/src/ModStdQt.jl b/experimental/ModStd/src/ModStdQt.jl
index b4268bc519bd..20f36b48a9e1 100644
--- a/experimental/ModStd/src/ModStdQt.jl
+++ b/experimental/ModStd/src/ModStdQt.jl
@@ -591,7 +591,7 @@ julia> f = factor_absolute((X[1]^2+a[1]*X[2]^2)*(X[1]+2*X[2]+3*a[1]+4*a[2]))
 
 julia> parent(f[3][1])
 Multivariate polynomial ring in 2 variables X[1], X[2]
-  over fraction field of multivariate polynomial ring
+  over fraction field of Qa
 
 julia> parent(f[2][1])
 Multivariate polynomial ring in 2 variables X[1], X[2]
diff --git a/experimental/Schemes/CoveredProjectiveSchemes.jl b/experimental/Schemes/CoveredProjectiveSchemes.jl
index 4a17ec3edcf3..6b56b606d78d 100644
--- a/experimental/Schemes/CoveredProjectiveSchemes.jl
+++ b/experimental/Schemes/CoveredProjectiveSchemes.jl
@@ -311,7 +311,7 @@ julia> R, (x,y,z) = QQ["x", "y", "z"];
 
 julia> Oscar.empty_covered_projective_scheme(R)
 Relative projective scheme
-  over empty covered scheme over multivariate polynomial ring
+  over empty covered scheme over R
 covered with 0 projective patches
 ```
 """
diff --git a/experimental/Schemes/duValSing.jl b/experimental/Schemes/duValSing.jl
index a359008e96ee..38e742e8ed5a 100644
--- a/experimental/Schemes/duValSing.jl
+++ b/experimental/Schemes/duValSing.jl
@@ -14,7 +14,7 @@ Ideal generated by
   x^2 + y^3 + z^4
 
 julia> Rq, _ = quo(R,I)
-(Quotient of multivariate polynomial ring by ideal (w, x^2 + y^3 + z^4), Map: multivariate polynomial ring -> Rq)
+(Quotient of multivariate polynomial ring by ideal (w, x^2 + y^3 + z^4), Map: R -> Rq)
 
 julia> X = spec(Rq)
 Spectrum
@@ -78,7 +78,7 @@ Ideal generated by
   x^2 + y^3 + z^4
 
 julia> Rq, _ = quo(R,I)
-(Quotient of multivariate polynomial ring by ideal (w, x^2 + y^3 + z^4), Map: multivariate polynomial ring -> Rq)
+(Quotient of multivariate polynomial ring by ideal (w, x^2 + y^3 + z^4), Map: R -> Rq)
 
 julia> J = ideal(R,[x,y,z,w])
 Ideal generated by
@@ -158,7 +158,7 @@ Ideal generated by
   x^2 + y^3 + z^4
 
 julia> Rq, _ = quo(R,I)
-(Quotient of multivariate polynomial ring by ideal (w, x^2 + y^3 + z^4), Map: multivariate polynomial ring -> Rq)
+(Quotient of multivariate polynomial ring by ideal (w, x^2 + y^3 + z^4), Map: R -> Rq)
 
 julia> J = ideal(R,[x,y,z,w])
 Ideal generated by
diff --git a/src/Combinatorics/SimplicialComplexes.jl b/src/Combinatorics/SimplicialComplexes.jl
index 1edd796bcb98..e00d6a293045 100644
--- a/src/Combinatorics/SimplicialComplexes.jl
+++ b/src/Combinatorics/SimplicialComplexes.jl
@@ -370,7 +370,7 @@ Return the Stanley-Reisner ring of the abstract simplicial complex `K`, as a quo
 julia>  R, _ = ZZ["a","b","c","d","e","f"];
 
 julia> stanley_reisner_ring(R, real_projective_plane())
-(Quotient of multivariate polynomial ring by ideal (a*b*c, a*b*d, a*e*f, b*e*f, a*c*f, a*d*e, c*d*e, c*d*f, b*c*e, b*d*f), Map: multivariate polynomial ring -> quotient of multivariate polynomial ring)
+(Quotient of multivariate polynomial ring by ideal (a*b*c, a*b*d, a*e*f, b*e*f, a*c*f, a*d*e, c*d*e, c*d*f, b*c*e, b*d*f), Map: R -> quotient of multivariate polynomial ring)
 ```
 """
 stanley_reisner_ring(R::MPolyRing, K::SimplicialComplex) = quo(R, stanley_reisner_ideal(R, K))
diff --git a/src/Groups/matrices/MatGrp.jl b/src/Groups/matrices/MatGrp.jl
index ce833179a1b1..439147c289a6 100644
--- a/src/Groups/matrices/MatGrp.jl
+++ b/src/Groups/matrices/MatGrp.jl
@@ -450,6 +450,8 @@ Return the base ring of the underlying matrix of `x`.
 """
 base_ring(x::MatrixGroupElem) = x.parent.ring
 
+base_ring_type(::Type{<:MatrixGroupElem{RE}}) where {RE} = parent_type(RE)
+
 parent(x::MatrixGroupElem) = x.parent
 
 """
@@ -512,6 +514,8 @@ Return the base ring of the matrix group `G`.
 """
 base_ring(G::MatrixGroup{RE}) where RE <: RingElem = G.ring::parent_type(RE)
 
+base_ring_type(::Type{<:MatrixGroup{RE}}) where {RE} = parent_type(RE)
+
 """
     degree(G::MatrixGroup)
 
diff --git a/src/Modules/ModuleTypes.jl b/src/Modules/ModuleTypes.jl
index a2416d45c949..b771dfac76a2 100644
--- a/src/Modules/ModuleTypes.jl
+++ b/src/Modules/ModuleTypes.jl
@@ -116,7 +116,7 @@ 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 R
 
 julia> f = F(sparse_row(R, [(1,x),(3,y)]))
 x*e[1] + y*e[3]
diff --git a/src/Modules/ModulesGraded.jl b/src/Modules/ModulesGraded.jl
index 01257946083d..ad82e60c8d40 100644
--- a/src/Modules/ModulesGraded.jl
+++ b/src/Modules/ModulesGraded.jl
@@ -2153,6 +2153,8 @@ Return the underlying ring of `F`.
 """
 base_ring(F::FreeMod_dec) = forget_decoration(F).R
 
+base_ring_type(::Type{FreeMod_dec{T}}) where {T} = base_ring_type(FreeMod{T})
+
 @doc raw"""
     rank(F::FreeMod_dec)
 
diff --git a/src/Modules/UngradedModules/FreeMod.jl b/src/Modules/UngradedModules/FreeMod.jl
index 86c3a881f3f5..e040f69abb42 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 R
 
 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 Localization of R at complement of prime ideal (x, y, z)
 
 julia> RL(x)*FRL[1]
 x*f[1]
diff --git a/src/Modules/UngradedModules/FreeModElem.jl b/src/Modules/UngradedModules/FreeModElem.jl
index ea29e58f9ba1..70e67c8d331c 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 R
 
 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 7053cb615649..569df691993a 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 R
 
 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 R
 
 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 R
 
 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 R
 
 julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]]
 3-element Vector{FreeModElem{QQMPolyRingElem}}:
@@ -116,10 +116,10 @@ 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
+Free module of rank 3 over R
 Codomain:
 =========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over R
 
 julia> a(F[2])
 x*e[1] + y*e[2]
@@ -133,10 +133,10 @@ 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
+Free module of rank 3 over R
 Codomain:
 =========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over R
 
 julia> a == b
 true
@@ -325,10 +325,10 @@ 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 R
 
 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 R
 
 julia> V, f = hom(F1, F2)
 (hom of (F1, F2), Map: V -> set of all homomorphisms from F1 to F2)
@@ -337,10 +337,10 @@ julia> f(V[1])
 Map with following data
 Domain:
 =======
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 3 over R
 Codomain:
 =========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over R
 
 ```
 
@@ -414,10 +414,10 @@ 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 R
 
 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 R
 
 julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]];
 
@@ -434,7 +434,7 @@ Submodule with 1 generator
 represented as subquotient with no relations.
 Codomain:
 =========
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ)
+Free module of rank 3 over R)
 ```
 
 ```jldoctest
@@ -546,10 +546,10 @@ 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 R
 
 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 R
 
 julia> V = [y*G[1], x*G[1]+y*G[2], z*G[2]];
 
@@ -570,7 +570,7 @@ Submodule with 3 generators
 represented as subquotient with no relations.
 Codomain:
 =========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ)
+Free module of rank 2 over R)
 ```
 
 ```jldoctest
diff --git a/src/Modules/UngradedModules/FreeResolutions.jl b/src/Modules/UngradedModules/FreeResolutions.jl
index de1e53c24f87..672d8c4696e4 100644
--- a/src/Modules/UngradedModules/FreeResolutions.jl
+++ b/src/Modules/UngradedModules/FreeResolutions.jl
@@ -284,7 +284,7 @@ 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 R
 
 julia> fr
 Free resolution of M
diff --git a/src/Modules/UngradedModules/ModuleGens.jl b/src/Modules/UngradedModules/ModuleGens.jl
index 02521bb98f10..f08661768b94 100644
--- a/src/Modules/UngradedModules/ModuleGens.jl
+++ b/src/Modules/UngradedModules/ModuleGens.jl
@@ -88,6 +88,8 @@ function base_ring(M::ModuleGens)
   return base_ring(M.F)
 end
 
+base_ring_type(::Type{ModuleGens{T}}) where {T} = base_ring_type(FreeMod{T})
+
 @doc raw"""
     singular_generators(M::ModuleGens)
 
diff --git a/src/Modules/UngradedModules/Presentation.jl b/src/Modules/UngradedModules/Presentation.jl
index c30a9d92f9f1..9bf8031f91c1 100644
--- a/src/Modules/UngradedModules/Presentation.jl
+++ b/src/Modules/UngradedModules/Presentation.jl
@@ -432,7 +432,7 @@ 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 R
 
 julia> present_as_cokernel(F)
 Submodule with 2 generators
@@ -444,7 +444,7 @@ 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
+Free module of rank 2 over R
 Codomain:
 =========
 Submodule with 2 generators
diff --git a/src/Modules/UngradedModules/SubModuleOfFreeModule.jl b/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
index 8a4e603433cb..ca8ae8494564 100644
--- a/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
+++ b/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
@@ -125,6 +125,9 @@ function base_ring(M::SubModuleOfFreeModule)
   return base_ring(M.F)
 end
 
+base_ring_type(::Type{SubModuleOfFreeModule{T}}) where {T} = base_ring_type(FreeMod{T})
+
+
 @doc raw"""
     ambient_free_module(M::SubModuleOfFreeModule)
 
diff --git a/src/Modules/UngradedModules/SubQuoHom.jl b/src/Modules/UngradedModules/SubQuoHom.jl
index 54b833e5e4a3..bfa2221091c9 100644
--- a/src/Modules/UngradedModules/SubQuoHom.jl
+++ b/src/Modules/UngradedModules/SubQuoHom.jl
@@ -121,7 +121,7 @@ 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 R
 
 julia> A = R[x; y]
 [x]
@@ -187,7 +187,7 @@ 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 R
 
 julia> A = R[x; y];
 
@@ -383,7 +383,7 @@ 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 R
 
 julia> A = R[x; y]
 [x]
@@ -602,7 +602,7 @@ Submodule with 3 generators
 represented as subquotient with no relations.
 Codomain:
 =========
-Free module of rank 2 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 2 over R
 ```
 
 ```jldoctest
@@ -780,7 +780,7 @@ Submodule with 1 generator
 represented as subquotient with no relations.
 Codomain:
 =========
-Free module of rank 3 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 3 over R
 ```
 
 ```jldoctest
diff --git a/src/Modules/UngradedModules/SubquoModule.jl b/src/Modules/UngradedModules/SubquoModule.jl
index 83821631c1d2..8c66a77c44a6 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 R
 
 julia> O = [x*F[1]+F[2],y*F[2]]
 2-element Vector{FreeModElem{QQMPolyRingElem}}:
@@ -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 R
 
 julia> AR = R[x; y]
 [x]
@@ -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 Localization of R at complement of prime ideal (x, y, z)
 
 julia> ARL = RL[x; y]
 [x]
@@ -571,7 +571,7 @@ 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 R
 
 julia> A = R[x y; 2*x^2 3*y^2]
 [    x       y]
@@ -658,7 +658,7 @@ 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 R
 
 julia> A = R[x y; 2*x^2 3*y^2]
 [    x       y]
@@ -838,7 +838,7 @@ 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 R
 
 julia> AM = R[x;]
 [x]
@@ -966,7 +966,7 @@ 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 R
 
 julia> AM = R[x;]
 [x]
@@ -1165,7 +1165,7 @@ 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 R
 
 julia> AM = R[x;]
 [x]
@@ -1347,7 +1347,7 @@ 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 R
 
 julia> AM = R[x;]
 [x]
@@ -1450,7 +1450,7 @@ 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 R
 
 julia> AM = R[x;]
 [x]
diff --git a/src/Modules/UngradedModules/SubquoModuleElem.jl b/src/Modules/UngradedModules/SubquoModuleElem.jl
index b59fdf355817..5e25c8d06ab1 100644
--- a/src/Modules/UngradedModules/SubquoModuleElem.jl
+++ b/src/Modules/UngradedModules/SubquoModuleElem.jl
@@ -624,7 +624,7 @@ Submodule with 3 generators
 represented as subquotient with no relations.
 Codomain:
 =========
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over R
 ```
 """
 function sub(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}; cache_morphism::Bool=false) where T
@@ -822,7 +822,7 @@ julia> proj
 Map with following data
 Domain:
 =======
-Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ
+Free module of rank 1 over R
 Codomain:
 =========
 Subquotient of Submodule with 1 generator
@@ -962,7 +962,9 @@ number_of_generators(M::SubquoModule) = number_of_generators(M.sub)
 
 Given an `R`-module `M`, return `R`.
 """
-base_ring(M::SubquoModule) = base_ring(M.F)::base_ring_type(M.F)
+base_ring(M::SubquoModule) = base_ring(M.F)::base_ring_type(M)
+
+base_ring_type(::Type{SubquoModule{T}}) where {T} = base_ring_type(FreeMod{T})
 
 @doc raw"""
     zero(M::SubquoModule)
@@ -982,7 +984,7 @@ 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 R
 
 julia> A = R[x^2+y^2;]
 [x^2 + y^2]
@@ -1038,7 +1040,7 @@ 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 R
 
 julia> A = R[x; y]
 [x]
diff --git a/src/PolyhedralGeometry/helpers.jl b/src/PolyhedralGeometry/helpers.jl
index 1733a098ce42..1a223840d881 100644
--- a/src/PolyhedralGeometry/helpers.jl
+++ b/src/PolyhedralGeometry/helpers.jl
@@ -216,6 +216,10 @@ end
 (F::Field)(x::Polymake.Rational) = F(QQ(x))
 (F::Field)(x::Polymake.OscarNumber) = F(Polymake.unwrap(x))
 
+# Disambiguation
+(F::QQBarField)(x::Polymake.Rational) = F(QQ(x))
+(F::QQBarField)(x::Polymake.OscarNumber) = F(Polymake.unwrap(x))
+
 Polymake.convert_to_pm_type(::Type{typeof(min)}) = Polymake.Min
 Polymake.convert_to_pm_type(::Type{typeof(max)}) = Polymake.Max
 
diff --git a/src/Rings/AbelianClosure.jl b/src/Rings/AbelianClosure.jl
index ee92f85bb71b..63317534d07f 100644
--- a/src/Rings/AbelianClosure.jl
+++ b/src/Rings/AbelianClosure.jl
@@ -34,7 +34,7 @@ import Base: +, *, -, //, ==, zero, one, ^, div, isone, iszero,
 
 #import ..Oscar.AbstractAlgebra: promote_rule
 
-import ..Oscar: AbstractAlgebra, addeq!, characteristic, elem_type, divexact, gen,
+import ..Oscar: AbstractAlgebra, addeq!, base_ring, base_ring_type, characteristic, elem_type, divexact, gen,
                 has_preimage_with_preimage, is_root_of_unity, is_unit, mul!, parent,
                 parent_type, promote_rule, root, root_of_unity, roots
 
@@ -151,6 +151,9 @@ elem_type(::Type{QQAbField{AbsNonSimpleNumField}}) = QQAbElem{AbsNonSimpleNumFie
 parent_type(::Type{QQAbElem{AbsNonSimpleNumFieldElem}}) = QQAbField{AbsNonSimpleNumField}
 parent(::QQAbElem{AbsNonSimpleNumFieldElem}) = _QQAb_sparse
 
+base_ring(::QQAbField) = Union{}
+base_ring_type(::Type{<:QQAbField}) = typeof(Union{})
+
 ################################################################################
 #
 #  Field access
diff --git a/src/Rings/AlgClosureFp.jl b/src/Rings/AlgClosureFp.jl
index 38bf0aaaeeeb..32c57802f697 100644
--- a/src/Rings/AlgClosureFp.jl
+++ b/src/Rings/AlgClosureFp.jl
@@ -16,7 +16,7 @@ import Base: +, -, *, //, ==, deepcopy_internal, hash, isone, iszero, one,
 
 import ..Oscar.AbstractAlgebra: pretty, Lowercase
 
-import ..Oscar: algebraic_closure, base_field, base_ring, characteristic, data, degree, divexact,
+import ..Oscar: algebraic_closure, base_field, base_ring, base_ring_type, characteristic, data, degree, divexact,
   elem_type, embedding, has_preimage_with_preimage, IntegerUnion, is_unit, map_entries,
   minpoly, parent_type, promote_rule, roots
 
@@ -36,6 +36,7 @@ end
 
 base_field(A::AlgClosure) = A.k
 base_ring(A::AlgClosure) = A.k
+base_ring_type(::Type{AlgClosure{T}}) where {T} = T
 characteristic(k::AlgClosure) = characteristic(base_field(k))
 
 struct AlgClosureElem{T} <: FieldElem
diff --git a/src/Rings/FractionalIdeal.jl b/src/Rings/FractionalIdeal.jl
index 7a948bd0fdf9..4b9fa5242067 100644
--- a/src/Rings/FractionalIdeal.jl
+++ b/src/Rings/FractionalIdeal.jl
@@ -30,6 +30,8 @@ function base_ring(a::FractionalIdeal)
   return base_ring(numerator(a))
 end
 
+base_ring_type(::Type{FractionalIdeal{S, T}}) where {S, T} = base_ring_type(S)
+
 function iszero(a::FractionalIdeal)
   return iszero(numerator(a))
 end
diff --git a/src/Rings/FreeAssAlgIdeal.jl b/src/Rings/FreeAssAlgIdeal.jl
index 8c636beb7ec4..ce21d0ff3505 100644
--- a/src/Rings/FreeAssAlgIdeal.jl
+++ b/src/Rings/FreeAssAlgIdeal.jl
@@ -51,6 +51,10 @@ function base_ring(I::FreeAssAlgIdeal{T}) where T
   return I.gens.Ox::parent_type(T)
 end
 
+function base_ring_type(::Type{<:FreeAssAlgIdeal{T}}) where T
+  return parent_type(T)
+end
+
 function number_of_generators(a::FreeAssAlgIdeal)
   return length(a.gens)
 end
diff --git a/src/Rings/Laurent.jl b/src/Rings/Laurent.jl
index 993ffefc1a0c..86051757d6f3 100644
--- a/src/Rings/Laurent.jl
+++ b/src/Rings/Laurent.jl
@@ -208,6 +208,8 @@ end
 
 base_ring(I::LaurentMPolyIdeal{T}) where {T} = I.R::parent_type(T)
 
+base_ring_type(::Type{LaurentMPolyIdeal{T}}) where {T} = parent_type(T)
+
 gens(I::LaurentMPolyIdeal) = I.gens
 
 @enable_all_show_via_expressify LaurentMPolyIdeal
diff --git a/src/Rings/MPolyMap/MPolyRing.jl b/src/Rings/MPolyMap/MPolyRing.jl
index e8135a7f8c73..fd9ac903c1d6 100644
--- a/src/Rings/MPolyMap/MPolyRing.jl
+++ b/src/Rings/MPolyMap/MPolyRing.jl
@@ -53,8 +53,8 @@ julia> R, (x, y) = polynomial_ring(K, ["x", "y"]);
 
 julia> F = hom(R, R, z -> z^2, [y, x])
 Ring homomorphism
-  from multivariate polynomial ring in 2 variables over GF(2, 2)
-  to multivariate polynomial ring in 2 variables over GF(2, 2)
+  from multivariate polynomial ring in 2 variables over K
+  to multivariate polynomial ring in 2 variables over K
 defined by
   x -> y
   y -> x
diff --git a/src/Rings/MPolyQuo.jl b/src/Rings/MPolyQuo.jl
index 2bc4ef53bb49..24ad06baf7ca 100644
--- a/src/Rings/MPolyQuo.jl
+++ b/src/Rings/MPolyQuo.jl
@@ -65,6 +65,7 @@ gens(Q::MPolyQuoRing) = [Q(x) for x = gens(base_ring(Q))]
 number_of_generators(Q::MPolyQuoRing) = number_of_generators(base_ring(Q))
 gen(Q::MPolyQuoRing, i::Int) = Q(gen(base_ring(Q), i))
 base_ring(Q::MPolyQuoRing) = base_ring(Q.I)
+base_ring_type(::Type{MPolyQuoRing{S}}) where {S} = base_ring_type(S)
 coefficient_ring(Q::MPolyQuoRing) = coefficient_ring(base_ring(Q))
 modulus(Q::MPolyQuoRing) = Q.I
 oscar_groebner_basis(Q::MPolyQuoRing) = _groebner_basis(Q) && return Q.I.gb[Q.ordering].O
@@ -331,7 +332,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]))
-(Quotient of multivariate polynomial ring by ideal (-x^2 + y, -x^3 + z), Map: multivariate polynomial ring -> A)
+(Quotient of multivariate polynomial ring by ideal (-x^2 + y, -x^3 + z), Map: R -> A)
 
 julia> a = ideal(A, [x-y])
 Ideal generated by
@@ -360,7 +361,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]))
-(Quotient of multivariate polynomial ring by ideal (-x^2 + y, -x^3 + z), Map: multivariate polynomial ring -> A)
+(Quotient of multivariate polynomial ring by ideal (-x^2 + y, -x^3 + z), Map: R -> A)
 
 julia> a = ideal(A, [x-y])
 Ideal generated by
@@ -1594,8 +1595,7 @@ julia> HC = gens(L[1]);
 
 julia> EMB = L[2]
 Map defined by a julia-function with inverse
-  from quotient space over:
-  Rational field with 7 generators and no relations
+  from quotient space over QQ with 7 generators and no relations
   to quotient of multivariate polynomial ring by ideal (-x*z + y^2, -w*z + x*y, -w*y + x^2)
 
 julia> for i in 1:length(HC) println(EMB(HC[i])) end
@@ -1646,8 +1646,7 @@ julia> HC = gens(L[1]);
 
 julia> EMB = L[2]
 Map defined by a julia-function with inverse
-  from quotient space over:
-  Rational field with 7 generators and no relations
+  from quotient space over QQ with 7 generators and no relations
   to quotient of multivariate polynomial ring by ideal (x[1]*y[1] - x[2]*y[2])
 
 julia> for i in 1:length(HC) println(EMB(HC[i])) end
diff --git a/src/Rings/PBWAlgebra.jl b/src/Rings/PBWAlgebra.jl
index d7f9d7afd4a4..756737efcab4 100644
--- a/src/Rings/PBWAlgebra.jl
+++ b/src/Rings/PBWAlgebra.jl
@@ -599,6 +599,8 @@ function base_ring(a::PBWAlgIdeal)
   return a.basering
 end
 
+base_ring_type(::Type{PBWAlgIdeal{D, T, S}}) where {D, T, S} = PBWAlgRing{T, S}
+
 function number_of_generators(a::PBWAlgIdeal)
   return number_of_generators(a.sdata)
 end
diff --git a/src/Rings/PBWAlgebraQuo.jl b/src/Rings/PBWAlgebraQuo.jl
index 60dcdf9288e6..72204fdcffcc 100644
--- a/src/Rings/PBWAlgebraQuo.jl
+++ b/src/Rings/PBWAlgebraQuo.jl
@@ -77,7 +77,9 @@ coefficient_ring(a::PBWAlgQuoElem) = coefficient_ring(parent(a))
 
 modulus(Q::PBWAlgQuo) = Q.I
 
-base_ring(Q::PBWAlgQuo) = Q.I.basering
+base_ring(Q::PBWAlgQuo) = base_ring(Q.I)
+
+base_ring_type(::Type{PBWAlgQuo{T, S}}) where {T, S} = base_ring_type(PBWAlgIdeal{0, T, S})
 
 base_ring(a::PBWAlgQuoElem) = base_ring(parent(a))
 
diff --git a/src/Rings/lazypolys.jl b/src/Rings/lazypolys.jl
index ec13921ffdb4..1914d9a1bf3d 100644
--- a/src/Rings/lazypolys.jl
+++ b/src/Rings/lazypolys.jl
@@ -11,6 +11,7 @@ end
 
 base_ring(F::LazyPolyRing) = F.base_ring
 
+base_ring_type(::Type{LazyPolyRing{T, R}}) where {T,R} = R
 
 ## LazyPoly
 
diff --git a/src/Rings/mpoly-graded.jl b/src/Rings/mpoly-graded.jl
index cf70e5d3ee56..42e0267e23f7 100644
--- a/src/Rings/mpoly-graded.jl
+++ b/src/Rings/mpoly-graded.jl
@@ -1169,6 +1169,7 @@ function homogeneous_component(a::MPolyDecRingElem, g::Vector{<:IntegerUnion})
 end
 
 base_ring(W::MPolyDecRing) = base_ring(forget_decoration(W))
+base_ring_type(::Type{MPolyDecRing{T, S}}) where {T, S} = base_ring_type(S)
 number_of_generators(W::MPolyDecRing) = number_of_generators(forget_decoration(W))
 gens(W::MPolyDecRing) = map(W, gens(forget_decoration(W)))
 gen(W::MPolyDecRing, i::Int) = W(gen(forget_decoration(W), i))
diff --git a/src/Rings/mpoly-ideals.jl b/src/Rings/mpoly-ideals.jl
index 2974e47520ae..3d07571daa76 100644
--- a/src/Rings/mpoly-ideals.jl
+++ b/src/Rings/mpoly-ideals.jl
@@ -1651,6 +1651,8 @@ function base_ring(I::MPolyIdeal{S}) where {S}
   return I.gens.Ox::parent_type(S)
 end
 
+base_ring_type(::Type{MPolyIdeal{S}}) where {S} = parent_type(S)
+
 #######################################################
 @doc raw"""
     coefficient_ring(I::MPolyIdeal)
diff --git a/src/Rings/mpoly-local.jl b/src/Rings/mpoly-local.jl
index d6127e138252..83acdcbf6328 100644
--- a/src/Rings/mpoly-local.jl
+++ b/src/Rings/mpoly-local.jl
@@ -96,6 +96,7 @@ function Base.show(io::IO, w::MPolyRingElemLoc)
 end
 
 base_ring(R::MPolyRingLoc) = R.base_ring
+base_ring_type(::Type{MPolyRingLoc{T}}) where {T} = MPolyRing{T}
 symbols(R::MPolyRingLoc) = symbols(base_ring(R))
 number_of_variables(R::MPolyRingLoc) = number_of_variables(base_ring(R))
 parent(f::MPolyRingElemLoc) = f.parent
diff --git a/src/Rings/mpoly-localizations.jl b/src/Rings/mpoly-localizations.jl
index d5107af41c27..c64ce0297e66 100644
--- a/src/Rings/mpoly-localizations.jl
+++ b/src/Rings/mpoly-localizations.jl
@@ -900,6 +900,7 @@ end
 
 ### required getter functions 
 base_ring(W::MPolyLocRing) = W.R
+base_ring_type(::Type{MPolyLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = BRT
 inverted_set(W::MPolyLocRing) = W.S
 
 ### additional getter functions
@@ -1308,6 +1309,7 @@ end
 gens(I::MPolyLocalizedIdeal) = copy(I.gens)
 gen(I::MPolyLocalizedIdeal, i::Int) = I.gens[i]
 base_ring(I::MPolyLocalizedIdeal) = I.W
+base_ring_type(::Type{MPolyLocalizedIdeal{LRT, LRET}}) where {LRT, LRET} = LRT
 
 ### type getters
 ideal_type(::Type{MPolyLocalizedRingType}) where {MPolyLocalizedRingType<:MPolyLocRing} = MPolyLocalizedIdeal{MPolyLocalizedRingType, elem_type(MPolyLocalizedRingType)}
@@ -1821,7 +1823,7 @@ Complement
   in multivariate polynomial ring in 2 variables over QQ
 
 julia> L, _ = localization(R, U)
-(Localization of multivariate polynomial ring in 2 variables over QQ at complement of prime ideal (x, y^2 + 1), Hom: multivariate polynomial ring -> localized ring)
+(Localization of multivariate polynomial ring in 2 variables over QQ at complement of prime ideal (x, y^2 + 1), Hom: R -> localized ring)
 
 julia> J = ideal(L,[y*(x^2+(y^2+1)^2)])
 Ideal generated by
diff --git a/src/Rings/mpolyquo-localizations.jl b/src/Rings/mpolyquo-localizations.jl
index 6cdac96bb474..3a1318d3336e 100644
--- a/src/Rings/mpolyquo-localizations.jl
+++ b/src/Rings/mpolyquo-localizations.jl
@@ -122,7 +122,6 @@ coefficient_ring_elem_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) whe
 coefficient_ring_elem_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = coefficient_ring_elem_type(typeof(L))
 
 base_ring_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = RT
-base_ring_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_type(typeof(L))
 base_ring_elem_type(::Type{MPolyQuoLocRing{BRT, BRET, RT, RET, MST}}) where {BRT, BRET, RT, RET, MST} = RET
 base_ring_elem_type(L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}) where {BRT, BRET, RT, RET, MST} = base_ring_elem_type(typeof(L))
 
@@ -1711,6 +1710,7 @@ end
 gens(I::MPolyQuoLocalizedIdeal) = copy(I.gens)
 gen(I::MPolyQuoLocalizedIdeal, i::Int) = I.gens[i]
 base_ring(I::MPolyQuoLocalizedIdeal) = I.W
+base_ring_type(::Type{MPolyQuoLocalizedIdeal{LRT, LET, MPT}}) where {LRT, LET, MPT} = LRT
 
 ### additional getter functions 
 map_from_base_ring(I::MPolyQuoLocalizedIdeal) = I.map_from_base_ring
diff --git a/src/Rings/orderings.jl b/src/Rings/orderings.jl
index 1eaba27347d9..746b67b60569 100644
--- a/src/Rings/orderings.jl
+++ b/src/Rings/orderings.jl
@@ -188,6 +188,8 @@ end
 
 base_ring(a::MonomialOrdering) = a.R
 
+base_ring_type(::Type{MonomialOrdering{S}}) where {S} = S
+
 @doc raw"""
     support(o::MonomialOrdering)
 
@@ -1569,7 +1571,7 @@ 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 R
 
 julia> cmp(lex(R)*invlex(F), F[1], F[2])
 -1
@@ -1770,6 +1772,8 @@ end
 
 base_ring(a::ModuleOrdering) = a.M
 
+base_ring_type(::Type{ModuleOrdering{S}}) where {S} = S
+
 mutable struct ModProdOrdering <: AbsModOrdering
   a::AbsOrdering
   b::AbsOrdering
@@ -1927,7 +1931,7 @@ 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 R
 
 julia> o = invlex(gens(F))*degrevlex(R)
 invlex([gen(1), gen(2), gen(3)])*degrevlex([w, x, y, z])
diff --git a/src/Rings/slpolys.jl b/src/Rings/slpolys.jl
index 4e890181790e..e5163dcb44ce 100644
--- a/src/Rings/slpolys.jl
+++ b/src/Rings/slpolys.jl
@@ -22,6 +22,8 @@ SLPolyRing(r::Ring, v::Pair{<:VarName,
 
 base_ring(S::SLPolyRing) = S.base_ring
 
+base_ring_type(::Type{SLPolyRing{T,R}}) where {T,R} = R
+
 elem_type(::Type{S}) where {T,S<:SLPolyRing{T}} = SLPoly{T,S}
 
 symbols(S::SLPolyRing) = S.S
diff --git a/src/aliases.jl b/src/aliases.jl
index 28534d5451e7..6df64e519069 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -1,7 +1,3 @@
-# HACK/FIXME: remove these aliases once https://github.com/Nemocas/AbstractAlgebra.jl/pull/1696 is available
-@alias characteristic_polynomial charpoly  # FIXME
-@alias minimal_polynomial minpoly  # FIXME
-
 # make some Julia names compatible with our naming conventions
 @alias is_subset issubset
 @alias is_valid isvalid
diff --git a/test/Rings/integer-localizations.jl b/test/Rings/integer-localizations.jl
index 212982071504..bac0559c4b41 100644
--- a/test/Rings/integer-localizations.jl
+++ b/test/Rings/integer-localizations.jl
@@ -4,7 +4,7 @@
 
 using Oscar
 
-import Oscar: base_ring, inverted_set, ring, localization, parent, numerator, denominator, one, zero, reduce_fraction
+import Oscar: base_ring, base_ring_type, inverted_set, ring, localization, parent, numerator, denominator, one, zero, reduce_fraction
 import Oscar.AbstractAlgebra: elem_type, parent_type
 
 export FmpzComplementOfPrimeIdeal, FmpzPowersOfElement, FmpzComplementOfZeroIdeal
@@ -139,6 +139,7 @@ end
 
 ### required getter functions
 base_ring(W::FmpzLocalizedRing) = ZZ::ZZRing
+base_ring_type(::Type{<:FmpzLocalizedRing}) = ZZRing
 inverted_set(W::FmpzLocalizedRing{MultSetType}) where {MultSetType} = W.S::MultSetType
 
 ### required extensions of the localization function
diff --git a/test/Rings/nmod-localizations.jl b/test/Rings/nmod-localizations.jl
index f677ec2c8164..1e9ed12c0847 100644
--- a/test/Rings/nmod-localizations.jl
+++ b/test/Rings/nmod-localizations.jl
@@ -5,7 +5,7 @@
 using Oscar
 
 import Oscar.Nemo.zzModRing
-import Oscar: base_ring, inverted_set, ring, localization, parent, numerator, denominator, one, zero
+import Oscar: base_ring, base_ring_type, inverted_set, ring, localization, parent, numerator, denominator, one, zero
 import Oscar.AbstractAlgebra: elem_type, parent_type
 
 export NmodComplementOfPrimeIdeal, NmodLocalizedRing, NmodLocalizedRingElem
@@ -71,6 +71,7 @@ end
 
 ### required getter functions
 base_ring(W::NmodLocalizedRing) = W.R::zzModRing
+base_ring_type(::Type{<:NmodLocalizedRing}) = zzModRing
 inverted_set(W::NmodLocalizedRing{MultSetType}) where {MultSetType} = W.S::MultSetType
 
 ### required extension of the localization function
diff --git a/test/book/cornerstones/groups/explSL25.jlcon b/test/book/cornerstones/groups/explSL25.jlcon
index bfe5f0582ff9..ed265be8f3b3 100644
--- a/test/book/cornerstones/groups/explSL25.jlcon
+++ b/test/book/cornerstones/groups/explSL25.jlcon
@@ -30,7 +30,7 @@ julia> schur_index(T[end])
 2
 
 julia> gmodule_minimal_field(S)
-G-module for G acting on vector space of dimension 6 over number field of degree 4 over QQ
+G-module for G acting on vector space of dimension 6 over number field
 
 julia> B, mB = relative_brauer_group(base_ring(S), character_field(S));
 
diff --git a/test/book/cornerstones/groups/intro.jlcon b/test/book/cornerstones/groups/intro.jlcon
index fb7e8a623aaf..8be4e4a235a1 100644
--- a/test/book/cornerstones/groups/intro.jlcon
+++ b/test/book/cornerstones/groups/intro.jlcon
@@ -63,7 +63,7 @@ julia> pts = collect(orb)
 julia> visualize(convex_hull(pts))
 
 julia> R2 = free_module(K, 2) # the "euclidean" plane over K
-Vector space of dimension 2 over field of algebraic numbers
+Vector space of dimension 2 over QQBar
 
 julia> A = R2([0,1])
 (Root 0 of x, Root 1.00000 of x - 1)
@@ -72,8 +72,8 @@ julia> pts = [A*mat_rot^i for i in 0:4];
 
 julia> sigma_1 = hom(R2, R2, [-R2[1], R2[2]])
 Module homomorphism
-  from vector space of dimension 2 over field of algebraic numbers
-  to vector space of dimension 2 over field of algebraic numbers
+  from vector space of dimension 2 over QQBar
+  to vector space of dimension 2 over QQBar
 
 julia> rot = hom(R2, R2, mat_rot);
 
diff --git a/test/book/cornerstones/groups/reps.jlcon b/test/book/cornerstones/groups/reps.jlcon
index 792bd17669d9..bf8736660733 100644
--- a/test/book/cornerstones/groups/reps.jlcon
+++ b/test/book/cornerstones/groups/reps.jlcon
@@ -12,9 +12,9 @@ julia> M = regular_gmodule(GF(7), G);
 
 julia> C = composition_factors_with_multiplicity(M)
 3-element Vector{Any}:
- (G-module for G acting on vector space of dimension 1 over prime field of characteristic 7, 1)
- (G-module for G acting on vector space of dimension 1 over prime field of characteristic 7, 1)
- (G-module for G acting on vector space of dimension 4 over prime field of characteristic 7, 2)
+ (G-module for G acting on vector space of dimension 1 over GF(7), 1)
+ (G-module for G acting on vector space of dimension 1 over GF(7), 1)
+ (G-module for G acting on vector space of dimension 4 over GF(7), 2)
 
 julia> [is_absolutely_irreducible(x[1]) for x in C]
 3-element Vector{Bool}:
@@ -28,12 +28,12 @@ Morphism of finite fields
   to finite field of degree 2 and characteristic 7
 
 julia> M = extension_of_scalars(C[3][1], phi)
-G-module for G acting on vector space of dimension 4 over finite field of degree 2 and characteristic 7
+G-module for G acting on vector space of dimension 4 over GF(7, 2)
 
 julia> composition_factors_with_multiplicity(M)
 2-element Vector{Any}:
- (G-module for G acting on vector space of dimension 2 over finite field of degree 2 and characteristic 7, 1)
- (G-module for G acting on vector space of dimension 2 over finite field of degree 2 and characteristic 7, 1)
+ (G-module for G acting on vector space of dimension 2 over GF(7, 2), 1)
+ (G-module for G acting on vector space of dimension 2 over GF(7, 2), 1)
 
 julia> G = pc_group(symmetric_group(4));
 
diff --git a/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon b/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon
index cfe2de4da68f..001284a2e2d1 100644
--- a/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon
+++ b/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon
@@ -2,7 +2,7 @@ julia> F, o = finite_field(7,2)
 (Finite field of degree 2 and characteristic 7, o)
 
 julia> R, (y,z) = F["y","z"]
-(Multivariate polynomial ring in 2 variables over GF(7, 2), FqMPolyRingElem[y, z])
+(Multivariate polynomial ring in 2 variables over F, FqMPolyRingElem[y, z])
 
 julia> save("p.mrdi", 2*y^3*z^4 + (o + 3)*z^2 + 5*o*y + 1)
 
diff --git a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon
index 4847f599b04a..2bcf95ed6cc7 100644
--- a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon
+++ b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon
@@ -2,7 +2,7 @@ julia> Kt,t = rational_function_field(QQ,"t")
 (Rational function field over QQ, t)
 
 julia> Kxy, (x,y) = Kt["x", "y"]
-(Multivariate polynomial ring in 2 variables over rational function field, AbstractAlgebra.Generic.MPoly{AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}}[x, y])
+(Multivariate polynomial ring in 2 variables over Kt, AbstractAlgebra.Generic.MPoly{AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}}[x, y])
 
 julia> f=-x^3-4*x^2+y^2+(-8*t^4)*x
 -x^3 - 4*x^2 - 8*t^4*x + y^2
diff --git a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon
index b5abf8cf3424..ca90d2698873 100644
--- a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon
+++ b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon
@@ -2,7 +2,7 @@ julia> S,(b2,b34,b4,b56,b6,b7)=polynomial_ring(QQ,["b2", "b34", "b4", "b56", "b6
 (Multivariate polynomial ring in 6 variables over QQ, QQMPolyRingElem[b2, b34, b4, b56, b6, b7])
 
 julia> R,(x, y, z)=polynomial_ring(S,["x", "y", "z"])
-(Multivariate polynomial ring in 3 variables over multivariate polynomial ring, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}[x, y, z])
+(Multivariate polynomial ring in 3 variables over S, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}[x, y, z])
 
 julia> g = y^2-x*(x-b2^2)*(x-(b34+b4)^2)*(x-(b4)^2)*(x-(b6+b56)^2)*(x-(b6)^2)*(x+(b7)^2);