diff --git a/experimental/MatroidRealizationSpaces/test/runtests.jl b/experimental/MatroidRealizationSpaces/test/runtests.jl
index 3f1fbe1653e8..08f30d49d9e0 100644
--- a/experimental/MatroidRealizationSpaces/test/runtests.jl
+++ b/experimental/MatroidRealizationSpaces/test/runtests.jl
@@ -119,7 +119,7 @@ end
   @test X isa AbsSpec
   @test !isdefined(X, :underlying_scheme)
   R = OO(X)
-  @test R isa MPolyQuoLocRing
+  @test R isa Oscar.MPolyQuoLocRing
   f = sum(gens(R))
   U = PrincipalOpenSubset(X, f)
   @test U isa AbsSpec
diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl
index 0290bf7b3632..3b772f28b047 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Attributes.jl
@@ -543,7 +543,7 @@ Spectrum
       over rational field
     by ideal(x^2 - 2*x*y + y^2)
 
-julia> U = MPolyComplementOfKPointIdeal(R,[0,0])
+julia> U = complement_of_point_ideal(R, [0,0])
 Complement
   of maximal ideal corresponding to rational point with coordinates (0, 0)
   in multivariate polynomial ring in 2 variables over QQ
@@ -646,7 +646,7 @@ julia> singular_locus(A3)
 julia> singular_locus(X)
 (V(x^2 - y^2 + z^2, z, y, x), Hom: V(x^2 - y^2 + z^2, z, y, x) -> V(x^2 - y^2 + z^2))
 
-julia> U = MPolyComplementOfKPointIdeal(R,[0,0,0])
+julia> U = complement_of_point_ideal(R, [0,0,0])
 Complement
   of maximal ideal corresponding to rational point with coordinates (0, 0, 0)
   in multivariate polynomial ring in 3 variables over QQ
diff --git a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Properties.jl b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Properties.jl
index 40f45bcdbe01..fdcabcd34dcd 100644
--- a/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Properties.jl
+++ b/src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Properties.jl
@@ -666,7 +666,7 @@ Spectrum
 julia> is_smooth(Y)
 false
 
-julia> U = MPolyComplementOfKPointIdeal(R,[1,1])
+julia> U = complement_of_point_ideal(R, [1,1])
 Complement
   of maximal ideal corresponding to rational point with coordinates (1, 1)
   in multivariate polynomial ring in 2 variables over QQ
diff --git a/src/Modules/Posur.jl b/src/Modules/Posur.jl
index d38661f25a9f..cc2e94a84d29 100644
--- a/src/Modules/Posur.jl
+++ b/src/Modules/Posur.jl
@@ -156,7 +156,7 @@ recommended to choose ``f`` to be the 'least complex' in
 an appropriate sense for ``R``.
 """
 function has_nonempty_intersection(U::AbsMultSet, I::Ideal)
-  R = ambient_ring(U)
+  R = ring(U)
   R == base_ring(I) || error("the multiplicative set and the ideal must be defined over the same ring")
   error("this method is not implemented for multiplicative sets of type $(typeof(U)) and ideals of type $(typeof(I)); see Posur: Linear systems over localizations of rings, arXiv:1709.08180v2, Definition 3.8 for the requirements of the implementation")
 end
@@ -201,7 +201,7 @@ function has_solution(
   }
   R = base_ring(A)
   R === base_ring(b) || error("matrices must be defined over the same ring")
-  R === ambient_ring(U) || error("multiplicative set must be defined over the same ring as the matrices")
+  R === ring(U) || error("multiplicative set must be defined over the same ring as the matrices")
   m = nrows(A)
   nrows(b) == 1 || error("can not solve for more than one row vector")
   n = ncols(A)
diff --git a/src/Modules/mpoly-localizations.jl b/src/Modules/mpoly-localizations.jl
index e6246c189079..644aeaaa88f0 100644
--- a/src/Modules/mpoly-localizations.jl
+++ b/src/Modules/mpoly-localizations.jl
@@ -10,7 +10,7 @@
 
 
 function has_nonempty_intersection(U::MPolyPowersOfElement, I::MPolyIdeal; check::Bool=true)
-  R = ambient_ring(U)
+  R = ring(U)
   R == base_ring(I) || error("the multiplicative set and the ideal must be defined over the same ring")
 
   d = prod(denominators(U); init=one(R))
@@ -22,7 +22,7 @@ function has_nonempty_intersection(U::MPolyPowersOfElement, I::MPolyIdeal; check
 end
 
 function has_nonempty_intersection(U::MPolyComplementOfPrimeIdeal, I::MPolyIdeal; check::Bool=true)
-  R = ambient_ring(U)
+  R = ring(U)
   R == base_ring(I) || error("the multiplicative set and the ideal must be defined over the same ring")
   P = prime_ideal(U)
   candidates = [(f, i) for (f, i) in zip(gens(I), 1:ngens(I)) if !(f in P)]
@@ -40,7 +40,7 @@ function has_nonempty_intersection(U::MPolyComplementOfPrimeIdeal, I::MPolyIdeal
 end
 
 function has_nonempty_intersection(U::MPolyComplementOfKPointIdeal, I::MPolyIdeal; check::Bool=true)
-  R = ambient_ring(U)
+  R = ring(U)
   R == base_ring(I) || error("the multiplicative set and the ideal must be defined over the same ring")
   a = point_coordinates(U)
   candidates = [(f, i) for (f, i) in zip(gens(I), 1:ngens(I)) if !(iszero(evaluate(f, a)))]
@@ -59,7 +59,7 @@ end
 
 function has_nonempty_intersection(U::MPolyProductOfMultSets, I::MPolyIdeal; check::Bool=true)
   J = I
-  R = ambient_ring(U) 
+  R = ring(U) 
   R == base_ring(I) || error("rings not compatible")
   Usets = sets(U)
   if length(Usets) == 1 
@@ -202,4 +202,4 @@ the shift map ``Φ : R → R`` which is moving the point of ``𝔪`` to the orig
   Mp_gens_shift = shift.(gens(Mp))
   result = SubModuleOfFreeModule(F_shifted,Mp_gens_shift)
   return result
-end
\ No newline at end of file
+end
diff --git a/src/Rings/localization_interface.jl b/src/Rings/localization_interface.jl
index bef27cf6874f..94200dd6d0fc 100644
--- a/src/Rings/localization_interface.jl
+++ b/src/Rings/localization_interface.jl
@@ -19,12 +19,12 @@ abstract type AbsMultSet{RingType<:Ring, RingElemType<:RingElem} end
 
 ### required getter functions
 @doc raw"""
-    ambient_ring(S::AbsMultSet)
+    ring(S::AbsMultSet)
 
 Return the ambient ring `R` for a multiplicatively closed set `S ⊂ R`.
 """
-function ambient_ring(S::AbsMultSet)
-  error("method `ambient_ring` not implemented for multiplicatively closed sets of type $(typeof(S))")
+function ring(S::AbsMultSet)
+  error("method `ring` not implemented for multiplicatively closed sets of type $(typeof(S))")
 end
 
 ### required functionality
@@ -41,7 +41,7 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 julia> P = ideal(R, [x])
 ideal(x)
 
-julia> U = MPolyComplementOfPrimeIdeal(P)
+julia> U = complement_of_prime_ideal(P)
 Complement
   of prime ideal(x)
   in multivariate polynomial ring in 3 variables over QQ
@@ -59,7 +59,7 @@ end
 # of the multiplicative set whenever possible. For instance, this is 
 # used to check well-definedness of homomorphisms from localized rings. 
 # By default, however, this iteration does nothing.
-Base.iterate(U::T) where {T<:AbsMultSet} = (one(ambient_ring(U)), 1)
+Base.iterate(U::T) where {T<:AbsMultSet} = (one(ring(U)), 1)
 Base.iterate(U::T, a::Tuple{<:RingElem, Int}) where {T<:AbsMultSet} = nothing
 Base.iterate(U::T, i::Int) where {T<:AbsMultSet} = nothing
 
@@ -98,13 +98,14 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 julia> P = ideal(R, [x])
 ideal(x)
 
-julia> U = MPolyComplementOfPrimeIdeal(P)
+julia> U = complement_of_prime_ideal(P)
 Complement
   of prime ideal(x)
   in multivariate polynomial ring in 3 variables over QQ
 
 julia> Rloc, _ = localization(U);
 
+
 julia> R === base_ring(Rloc)
 true
 ```
@@ -126,13 +127,14 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 julia> P = ideal(R, [x])
 ideal(x)
 
-julia> U = MPolyComplementOfPrimeIdeal(P)
+julia> U = complement_of_prime_ideal(P)
 Complement
   of prime ideal(x)
   in multivariate polynomial ring in 3 variables over QQ
 
 julia> Rloc, _ = localization(U);
 
+
 julia> U === inverted_set(Rloc)
 true
 ```
@@ -160,13 +162,14 @@ julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
 julia> P = ideal(R, [x])
 ideal(x)
 
-julia> U = MPolyComplementOfPrimeIdeal(P)
+julia> U = complement_of_prime_ideal(P)
 Complement
   of prime ideal(x)
   in multivariate polynomial ring in 3 variables over QQ
 
 julia> Rloc, iota = localization(R, U);
 
+
 julia> Rloc
 Localization
   of multivariate polynomial ring in 3 variables x, y, z
@@ -188,7 +191,7 @@ function localization(S::AbsMultSet)
 end
 
 function localization(R::Ring, U::AbsMultSet)
-  R == ambient_ring(U) || error("ring and multiplicative set are incompatible")
+  R == ring(U) || error("ring and multiplicative set are incompatible")
   return localization(U)
 end
 
diff --git a/src/Rings/mpoly-localizations.jl b/src/Rings/mpoly-localizations.jl
index c83a01e4cf2d..fca7eb32c03f 100644
--- a/src/Rings/mpoly-localizations.jl
+++ b/src/Rings/mpoly-localizations.jl
@@ -70,7 +70,7 @@ mutable struct MPolyPowersOfElement{
 end
 
 ### required getter functions
-ambient_ring(S::MPolyPowersOfElement) = S.R
+ring(S::MPolyPowersOfElement) = S.R
 
 ### additional constructors
 MPolyPowersOfElement(f::RET) where {RET<:MPolyRingElem} = MPolyPowersOfElement(parent(f), [f])
@@ -85,7 +85,7 @@ function Base.in(
     S::MPolyPowersOfElement{BaseRingType, BaseRingElemType, RingType, RingElemType}
   ) where {BaseRingType, BaseRingElemType, RingType, RingElemType}
   R = parent(f)
-  R == ambient_ring(S) || return false
+  R === ring(S) || return false
   if iszero(f) 
     return false
   end
@@ -102,7 +102,7 @@ Base.iterate(U::MPolyPowersOfElement) = (length(U.a)>0 ? (U.a[1], 1) : nothing)
 Base.iterate(U::MPolyPowersOfElement, a::Tuple{<:MPolyRingElem, Int}) = (a[2] < length(U.a) ? (U.a[a[2]+1], a[2]+1) : nothing)
 Base.iterate(U::MPolyPowersOfElement, i::Int) = (i < length(U.a) ? (U.a[i+1], i+1) : nothing)
 
-is_trivial(U::MPolyPowersOfElement) = (U == units_of(ambient_ring(U)))
+is_trivial(U::MPolyPowersOfElement) = (U == units_of(ring(U)))
 
 ### printing
 
@@ -110,7 +110,7 @@ function Base.show(io::IO, ::MIME"text/plain", S::MPolyPowersOfElement)
   io = pretty(io)
   println(io, "Multiplicative subset")
   print(io, Indent())
-  println(io, "of ", Lowercase(), ambient_ring(S))
+  println(io, "of ", Lowercase(), ring(S))
   print(io, "given by the products of ")
   print(io, "[")
   join(io, denominators(S), ", ")
@@ -132,19 +132,19 @@ end
 
 ### generation of random elements 
 function rand(S::MPolyPowersOfElement, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  R = ambient_ring(S)
+  R = ring(S)
   return prod(f^rand(0:2) for f in denominators(S); init = one(R))::elem_type(R)
 end
 
 function rand(rng::Random.AbstractRNG, S::MPolyPowersOfElement, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  R = ambient_ring(S)
+  R = ring(S)
   return prod(f^rand(rng, 0:2) for f in denominators(S); init = one(R))::elem_type(R)
 end
 
 ### simplification. 
 # Replaces each element d by its list of square free prime divisors.
 function simplify!(S::MPolyPowersOfElement)
-  R = ambient_ring(S)
+  R = ring(S)
   new_denom = Vector{elem_type(R)}()
   for d in denominators(S)
     for a in factor(d)
@@ -156,7 +156,7 @@ function simplify!(S::MPolyPowersOfElement)
 end
 
 function simplify(S::MPolyPowersOfElement)
-  R = ambient_ring(S)
+  R = ring(S)
   new_denom = Vector{elem_type(R)}()
   for d in denominators(S)
     for a in factor(d)
@@ -228,8 +228,7 @@ mutable struct MPolyComplementOfPrimeIdeal{
 end
 
 ### required getter functions
-ambient_ring(
-    S::MPolyComplementOfPrimeIdeal) = S.R
+ring(S::MPolyComplementOfPrimeIdeal) = S.R
 
 ### required functionality
 function Base.in(
@@ -247,7 +246,7 @@ function Base.show(io::IO, ::MIME"text/plain", S::MPolyComplementOfPrimeIdeal)
   io = pretty(io)
   println(io, "Complement")
   println(io, Indent(), "of prime ", Lowercase(), prime_ideal(S))
-  print(io, "in ", Lowercase(), ambient_ring(S))
+  print(io, "in ", Lowercase(), ring(S))
   print(io, Dedent())
 end
 
@@ -261,7 +260,7 @@ end
 
 ### generation of random elements 
 function rand(S::MPolyComplementOfPrimeIdeal, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  f = rand(ambient_ring(S), v1, v2, v3)
+  f = rand(ring(S), v1, v2, v3)
   if f in prime_ideal(S)
     return rand(S, v1, v2, v3)
   end
@@ -269,7 +268,7 @@ function rand(S::MPolyComplementOfPrimeIdeal, v1::AbstractUnitRange{Int}, v2::Ab
 end
 
 function rand(rng::Random.AbstractRNG, S::MPolyComplementOfPrimeIdeal, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  f = rand(rng, ambient_ring(S), v1, v2, v3)
+  f = rand(rng, ring(S), v1, v2, v3)
   if f in prime_ideal(S)
     return rand(rng, S, v1, v2, v3)
   end
@@ -341,16 +340,16 @@ end
 # MPolyComplementOfPrimeIdeal. In order to treat them in 
 # parallel, we introduce the analogous method here. 
 function prime_ideal(S::MPolyComplementOfKPointIdeal) 
-  kk = coefficient_ring(ambient_ring(S))
+  kk = coefficient_ring(ring(S))
   # TODO: Test whether kk is an integral domain
   if !isdefined(S, :m)
-    R = ambient_ring(S)
+    R = ring(S)
     x = gens(R)
     n = ngens(R)
     m = ideal(R, [x[i]-a for (i, a) in enumerate(point_coordinates(S))])
     set_attribute!(m, :is_prime=>true)
     kk isa Field && set_attribute!(m, :is_maximal=>true)
-    set_attribute!(m, :dim=>dim(coefficient_ring(ambient_ring(S))))
+    set_attribute!(m, :dim=>dim(coefficient_ring(ring(S))))
     S.m = m
   end
   return S.m
@@ -430,7 +429,7 @@ Multiplicative subset
 powers_of_element(f::MPolyRingElem) = MPolyPowersOfElement(f)
 
 ### required getter functions
-ambient_ring(S::MPolyComplementOfKPointIdeal) = S.R
+ring(S::MPolyComplementOfKPointIdeal) = S.R
 
 ### additional getter functions 
 point_coordinates(S::MPolyComplementOfKPointIdeal) = S.a
@@ -440,8 +439,8 @@ function Base.in(
     f::RingElemType, 
     S::MPolyComplementOfKPointIdeal{BaseRingType, BaseRingElemType, RingType, RingElemType}
   ) where {BaseRingType, BaseRingElemType, RingType, RingElemType}
-  parent(f) == ambient_ring(S) || return false
-  return !(evaluate(f, point_coordinates(S)) == zero(ambient_ring(S)))
+  parent(f) == ring(S) || return false
+  return !(evaluate(f, point_coordinates(S)) == zero(ring(S)))
 end
 
 ### printing
@@ -454,7 +453,7 @@ function Base.show(io::IO, ::MIME"text/plain", S::MPolyComplementOfKPointIdeal)
   join(io, point_coordinates(S), ", ")
   println(io, ")")
   print(io, Dedent())
-  print(io, "in ", Lowercase(), ambient_ring(S))
+  print(io, "in ", Lowercase(), ring(S))
   print(io, Dedent())
 end
 
@@ -471,14 +470,14 @@ end
 
 ### generation of random elements 
 function rand(S::MPolyComplementOfKPointIdeal, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  f = rand(ambient_ring(S), v1, v2, v3)
+  f = rand(ring(S), v1, v2, v3)
   if !(f in S)
     return rand(S, v1, v2, v3)
   end
   return f
 end
 function rand(rng::Random.AbstractRNG, S::MPolyComplementOfKPointIdeal, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  f = rand(rng, ambient_ring(S), v1, v2, v3)
+  f = rand(rng, ring(S), v1, v2, v3)
   if !(f in S)
     return rand(rng, S, v1, v2, v3)
   end
@@ -509,7 +508,7 @@ julia> S = complement_of_point_ideal(R,[1,2]);
 
 julia> T = powers_of_element(x);
 
-julia> ST = MPolyProductOfMultSets(R, [S, T])
+julia> ST = Oscar.MPolyProductOfMultSets(R, [S, T])
 Product of the multiplicative sets
   complement of maximal ideal of point (1, 2)
   products of (x)
@@ -535,14 +534,14 @@ mutable struct MPolyProductOfMultSets{
 
   function MPolyProductOfMultSets(R::RT, U::Vector{<:AbsMPolyMultSet{BRT, BRET, RT, RET}}) where {BRT<:Ring, BRET<:RingElement, RT<:MPolyRing, RET<:MPolyRingElem}
     for s in U
-      ambient_ring(s) == R || error("multiplicative set does not live in the given ring")
+      ring(s) == R || error("multiplicative set does not live in the given ring")
     end
     return new{typeof(coefficient_ring(R)), elem_type(coefficient_ring(R)), typeof(R), elem_type(R)}(R, U)
   end
 end
 
 ### required getter functions
-ambient_ring(S::MPolyProductOfMultSets) = S.R
+ring(S::MPolyProductOfMultSets) = S.R
 
 ### additional functionality
 getindex(S::MPolyProductOfMultSets, i::Integer) = S.U[i]
@@ -553,7 +552,7 @@ function Base.in(
     f::RingElemType, 
     S::MPolyProductOfMultSets{BaseRingType, BaseRingElemType, RingType, RingElemType}
   ) where {BaseRingType, BaseRingElemType, RingType, RingElemType}
-  R = ambient_ring(S)
+  R = ring(S)
   divides(one(R), f)[1] && return true
   U = sets(S)
   for s in U
@@ -604,10 +603,10 @@ end
 
 ### generation of random elements 
 function rand(S::MPolyProductOfMultSets, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  return prod([rand(s, v1, v2, v3) for s in sets(S)])::elem_type(ambient_ring(S))
+  return prod([rand(s, v1, v2, v3) for s in sets(S)])::elem_type(ring(S))
 end
 function rand(rng::Random.AbstractRNG, S::MPolyProductOfMultSets, v1::AbstractUnitRange{Int}, v2::AbstractUnitRange{Int}, v3::AbstractUnitRange{Int})
-  return prod([rand(rng, s, v1, v2, v3) for s in sets(S)])::elem_type(ambient_ring(S))
+  return prod([rand(rng, s, v1, v2, v3) for s in sets(S)])::elem_type(ring(S))
 end
 
 ########################################################################
@@ -653,7 +652,7 @@ mutable struct MPolyLeadingMonOne{
 end
 
 ### required getter functions
-ambient_ring(S::MPolyLeadingMonOne) = S.R
+ring(S::MPolyLeadingMonOne) = S.R
 
 ### additional constructors
 MPolyLeadingMonOne(ord::MonomialOrdering) = MPolyLeadingMonOne(ord.R, ord)
@@ -666,7 +665,7 @@ function Base.in(
     S::MPolyLeadingMonOne{BaseRingType, BaseRingElemType, RingType, RingElemType}
   ) where {BaseRingType, BaseRingElemType, RingType, RingElemType}
   R = parent(f)
-  R == ambient_ring(S) || return false
+  R == ring(S) || return false
   if iszero(f)
     return false
   end
@@ -678,7 +677,7 @@ function Base.show(io::IO,::MIME"text/plain", S::MPolyLeadingMonOne)
   io = pretty(io)
   println(io, "Elements")
   print(io, Indent())
-  println(io, "of ", ambient_ring(S))
+  println(io, "of ", ring(S))
   print(io, Dedent())
   println(io, "with leading monomial 1 w.r.t. ")
   print(io, Indent())
@@ -704,7 +703,7 @@ end
 ⊂(T::AbsMPolyMultSet, U::AbsMPolyMultSet) = issubset(T, U)
 
 function issubset(T::AbsMPolyMultSet, U::AbsMPolyMultSet) 
-  ambient_ring(T) == ambient_ring(U) || return false
+  ring(T) == ring(U) || return false
   error("comparison of multiplicative sets of type $(typeof(T)) and $(typeof(U)) is not implemented")
 end
 
@@ -728,8 +727,8 @@ function issubset(
     T::MPolyComplementOfKPointIdeal{BRT, BRET, RT, RET},
     U::MPolyComplementOfKPointIdeal{BRT, BRET, RT, RET}
   ) where {BRT, BRET, RT, RET}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   a = point_coordinates(U)
   b = point_coordinates(T)
   for i in 1:length(a)
@@ -742,8 +741,8 @@ function issubset(
     T::MPolyComplementOfKPointIdeal{BRT, BRET, RT, RET},
     U::MPolyComplementOfPrimeIdeal{BRT, BRET, RT, RET}
   ) where {BRT, BRET, RT, RET}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   a = point_coordinates(T)
   for i in 1:length(a)
     (gen(R, i)- R(a[i])) in prime_ideal(U) || return false
@@ -755,8 +754,8 @@ function issubset(
     T::MPolyComplementOfPrimeIdeal{BRT, BRET, RT, RET},
     U::MPolyComplementOfKPointIdeal{BRT, BRET, RT, RET}
   ) where {BRT, BRET, RT, RET}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   a = point_coordinates(U)
   for f in gens(prime_ideal(T))
     iszero(evaluate(f, a)) || return false
@@ -897,16 +896,16 @@ Product of the multiplicative sets
 ```
 """
 function product(T::AbsMPolyMultSet, U::AbsMPolyMultSet)
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   issubset(T, U) && return U
   issubset(U, T) && return T
   return MPolyProductOfMultSets(R, [T, U])
 end
 
 function product(T::MST, U::MST) where {MST<:MPolyProductOfMultSets} 
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   new_sets = Vector()
   for S in sets(T)
     push!(new_sets, S)
@@ -931,8 +930,8 @@ function product(T::MST, U::MST) where {MST<:MPolyProductOfMultSets}
 end
 
 function product(T::MPolyProductOfMultSets{BRT, BRET, RT, RET}, U::MST) where {BRT, BRET, RT, RET, MST<:AbsMPolyMultSet{BRT, BRET, RT, RET}}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   for V in sets(T)
     issubset(U, T) && return T
   end
@@ -946,8 +945,8 @@ end
 product(U::MST, T::MPolyProductOfMultSets{BRT, BRET, RT, RET}) where {BRT, BRET, RT, RET, MST<:AbsMPolyMultSet{BRT, BRET, RT, RET}} = product(T, U)
 
 function product(T::MPolyComplementOfPrimeIdeal{BRT, BRET, RT, RET}, U::MPolyComplementOfKPointIdeal{BRT, BRET, RT, RET}) where {BRT, BRET, RT, RET}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   P = prime_ideal(T)
   for f in gens(P)
     if iszero(evaluate(f, point_coordinates(U)))
@@ -965,8 +964,8 @@ function product(
 end
 
 function product(T::MST, U::MST) where {MST<:MPolyComplementOfKPointIdeal}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   a = point_coordinates(U)
   b = point_coordinates(T)
   for i in 1:length(a)
@@ -976,8 +975,8 @@ function product(T::MST, U::MST) where {MST<:MPolyComplementOfKPointIdeal}
 end
 
 function product(T::MST, U::MST) where {MST<:MPolyComplementOfPrimeIdeal}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   if issubset(prime_ideal(T), prime_ideal(U))
     return T
   end
@@ -988,8 +987,8 @@ function product(T::MST, U::MST) where {MST<:MPolyComplementOfPrimeIdeal}
 end
 
 function product(T::MST, U::MST) where {MST<:MPolyPowersOfElement}
-  R = ambient_ring(T) 
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T) 
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   new_denoms = Vector{elem_type(R)}()
   for f in denominators(T)
     for g in denominators(U)
@@ -1015,8 +1014,8 @@ function product(
     U::AbsMPolyMultSet{BRT, BRET, RT, RET},
     T::MPolyPowersOfElement{BRT, BRET, RT, RET}
   ) where {BRT, BRET, RT, RET}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   for a in denominators(T)
     a in U || return MPolyProductOfMultSets(R, [U, T])
   end
@@ -1027,8 +1026,8 @@ function product(
     T::MPolyPowersOfElement{BRT, BRET, RT, RET},
     U::AbsMPolyMultSet{BRT, BRET, RT, RET}
   ) where {BRT, BRET, RT, RET}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   for a in denominators(T)
     a in U || return MPolyProductOfMultSets(R, [U, T])
   end
@@ -1039,8 +1038,8 @@ function product(
     T::MPolyPowersOfElement{BRT, BRET, RT, RET},
     U::MPolyProductOfMultSets{BRT, BRET, RT, RET}
   ) where {BRT, BRET, RT, RET}
-  R = ambient_ring(T)
-  R == ambient_ring(U) || error("multiplicative sets do not belong to the same ring")
+  R = ring(T)
+  R == ring(U) || error("multiplicative sets do not belong to the same ring")
   keep_denom = Vector{RET}()
   for a in denominators(T)
     a in U || (push!(keep_denom, a))
@@ -1070,7 +1069,7 @@ end
 function (phi::MPolyAnyMap{<:MPolyRing, <:MPolyRing, Nothing})(U::MPolyPowersOfElement;
                                                                check::Bool=true
                                                               )
-  ambient_ring(U) === domain(phi) || error("multiplicative set does not lay in the domain of the morphism")
+  ring(U) === domain(phi) || error("multiplicative set does not lay in the domain of the morphism")
   S = codomain(phi) 
   SU = MPolyPowersOfElement(S, phi.(denominators(U)))
   return SU
@@ -1079,7 +1078,7 @@ end
 function (phi::MPolyAnyMap{<:MPolyRing, <:MPolyRing, Nothing})(U::MPolyComplementOfPrimeIdeal;
                                                                check::Bool=true
                                                               )
-  ambient_ring(U) === domain(phi) || error("multiplicative set does not lay in the domain of the morphism")
+  ring(U) === domain(phi) || error("multiplicative set does not lay in the domain of the morphism")
   S = codomain(phi) 
   Q = ideal(S, phi.(gens(prime_ideal(U))))
   SU = MPolyComplementOfPrimeIdeal(S, Q, check=check)
@@ -1126,7 +1125,7 @@ at a multiplicative set ``S ⊂ R`` of type `MultSetType`.
       S::MultSetType
     ) where {RingType<:MPolyRing, MultSetType<:AbsMPolyMultSet}
     # TODO: Add some sanity checks here?
-    ambient_ring(S) == R || error("the multiplicative set is not contained in the given ring")
+    ring(S) == R || error("the multiplicative set is not contained in the given ring")
     k = coefficient_ring(R)
     R_loc = new{typeof(k), elem_type(k), RingType, elem_type(R), MultSetType}(R, S)
     return R_loc
@@ -1180,7 +1179,7 @@ defined by
 """ localization(R::MPolyRing, U::AbsMPolyMultSet)
 
 function localization(S::AbsMPolyMultSet)
-    R = ambient_ring(S)
+    R = ring(S)
     Rloc = MPolyLocRing(R, S)
     #iota = MapFromFunc(R, Rloc, x -> Rloc(x))
     iota = hom(R, Rloc, Rloc.(gens(R)), check=false)
@@ -1692,7 +1691,7 @@ julia> R,(x,y,z,w) = QQ["x","y","z","w"];
 
 julia> f = x+y+z+w-1;
 
-julia> T = MPolyPowersOfElement(f);
+julia> T = powers_of_element(f);
 
 julia> RL,phiL = localization(R,T);
 
@@ -2168,16 +2167,13 @@ julia> R, (x, y) = QQ["x", "y"]
 julia> I = ideal(R, [x, y^2+1])
 ideal(x, y^2 + 1)
 
-julia> U = MPolyComplementOfPrimeIdeal(I)
+julia> U = complement_of_prime_ideal(I)
 Complement
   of prime ideal(x, y^2 + 1)
   in multivariate polynomial ring in 2 variables over QQ
 
-julia> L = MPolyLocRing(R, U)
-Localization
-  of multivariate polynomial ring in 2 variables x, y
-    over rational field
-  at complement of prime ideal(x, y^2 + 1)
+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)
 
 julia> J = ideal(L,[y*(x^2+(y^2+1)^2)])
 Ideal
@@ -3112,7 +3108,7 @@ with 4 generators
   y^2 - 4*y + 4
 
 julia> minimal_generating_set(I)
-3-element Vector{MPolyLocRingElem{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem, MPolyComplementOfKPointIdeal{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem}}}:
+3-element Vector{Oscar.MPolyLocRingElem{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem, Oscar.MPolyComplementOfKPointIdeal{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem}}}:
  y^2 - 4*y + 4
  x*y - 2*x - y + 2
  x^2 - 2*x + 1
@@ -3188,7 +3184,7 @@ with 4 generators
   y^2
 
 julia> small_generating_set(I)
-3-element Vector{MPolyLocRingElem{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem, MPolyPowersOfElement{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem}}}:
+3-element Vector{Oscar.MPolyLocRingElem{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem, Oscar.MPolyPowersOfElement{QQField, QQFieldElem, QQMPolyRing, QQMPolyRingElem}}}:
  y^2
  x*y - y
  x^2 - 2*x + 1
diff --git a/src/Rings/mpolyquo-localizations.jl b/src/Rings/mpolyquo-localizations.jl
index 2ea292740008..9847a64a9182 100644
--- a/src/Rings/mpolyquo-localizations.jl
+++ b/src/Rings/mpolyquo-localizations.jl
@@ -98,7 +98,7 @@ multiplicative set ``S ⊂ P`` of type `MultSetType`.
     #modulus(Q) == I || error("the modulus of the quotient ring does not coincide with the ideal")
     S === inverted_set(W) || error("the multiplicative set does not coincide with the inverted set of the localized ring")
     base_ring(W) === R || error("the localization does not come from the given ring")
-    ambient_ring(S) === R || error("Multiplicative set does not belong to the ring")
+    ring(S) === R || error("Multiplicative set does not belong to the ring")
     k = coefficient_ring(R)
     L = new{typeof(k), elem_type(k), typeof(R), RingElemType, MultSetType}(R, I, S, Q, W)
     return L
@@ -324,7 +324,7 @@ function localization(
     L::MPolyQuoLocRing{BRT, BRET, RT, RET, MST}, 
     S::AbsMPolyMultSet{BRT, BRET, RT, RET}
   ) where {BRT, BRET, RT, RET, MST}
-  ambient_ring(S) === base_ring(L) || error("multiplicative set does not belong to the correct ring")
+  ring(S) === base_ring(L) || error("multiplicative set does not belong to the correct ring")
   issubset(S, inverted_set(L)) && return L, MapFromFunc(L, L, x->x)
   U = inverted_set(L)*S
   W = MPolyQuoLocRing(base_ring(L), modulus(underlying_quotient(L)), U, underlying_quotient(L), localization(U)[1])
@@ -1643,7 +1643,7 @@ julia> Q = ideal(R,[x*y-z*w]);
 
 julia> RQ,phiQ = quo(R,Q);
 
-julia> T = MPolyComplementOfKPointIdeal(R,[0,0,0,0]);
+julia> T = complement_of_point_ideal(R,[0,0,0,0]);
 
 julia> RQL, phiQL = localization(RQ,T);
 
diff --git a/src/aliases.jl b/src/aliases.jl
index 9fcbf61d50b7..41f753185705 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -104,3 +104,4 @@ Base.@deprecate_binding SubQuo SubquoModule
 #@alias SubQuoElem_dec SubquoDecModuleElem
 #@alias SubQuo_dec SubquoDecModule
 Base.@deprecate_binding GradedPolynomialRing graded_polynomial_ring
+
diff --git a/src/deprecations.jl b/src/deprecations.jl
index 61deabd1970c..f072c3e1878b 100644
--- a/src/deprecations.jl
+++ b/src/deprecations.jl
@@ -528,6 +528,7 @@ Base.@deprecate_binding is_isomorphic_with_alternating_group is_isomorphic_to_al
 Base.@deprecate_binding has_is_isomorphic_with_alternating_group has_is_isomorphic_to_alternating_group
 Base.@deprecate_binding set_is_isomorphic_with_alternating_group set_is_isomorphic_to_alternating_group
 
-# Deprecated after 0.15.0
+Base.@deprecate ambient_ring(U::AbsMultSet) ring(U)
+
 Base.@deprecate_binding jacobi_matrix jacobian_matrix
 Base.@deprecate_binding jacobi_ideal jacobian_ideal
diff --git a/src/exports.jl b/src/exports.jl
index 5d1d2bcdd4b7..875d0c0c0192 100644
--- a/src/exports.jl
+++ b/src/exports.jl
@@ -90,23 +90,10 @@ export LinearHalfspace
 export LinearHyperplane
 export LinearProgram, linear_program
 export localization
-export MPolyComplementOfKPointIdeal
-export MPolyComplementOfPrimeIdeal
 export MPolyDecRing
 export MPolyDecRingElem
 export MPolyIdeal
-export MPolyLeadingMonOne
-export MPolyLocRing
-export MPolyLocRingElem
-export MPolyLocalizedIdeal
-export MPolyLocalizedRingHom
-export MPolyPowersOfElement
-export MPolyProductOfMultSets
 export MPolyQuoIdeal
-export MPolyQuoLocRing
-export MPolyQuoLocRingElem
-export MPolyQuoLocalizedIdeal
-export MPolyQuoLocalizedRingHom
 export MPolyQuoRing
 export MPolyQuoRingElem
 export MPolyRingElem
@@ -246,7 +233,6 @@ export ambient_germ
 export ambient_module
 export ambient_representative
 export ambient_representatives_generators
-export ambient_ring
 export ambient_scheme
 export ambient_space
 export ambient_type
@@ -269,9 +255,6 @@ export augmented_chow_ring
 export aut
 export automorphism_group
 export base_ring
-export base_ring_elem_type
-export base_ring_module
-export base_ring_type
 export base_scheme
 export bases
 export basic_patches
@@ -294,7 +277,6 @@ export blocks
 export bond_matroid
 export borcherds_method
 export boundary_lattice_points
-export bring_to_common_denominator
 export build_ctx
 export build_doc
 export canonical_bundle
@@ -365,7 +347,6 @@ export combinatorial_symmetries
 export comm
 export comm!
 export common_components
-export common_denominator
 export common_refinement
 export complement
 export complement_class_reps
@@ -640,11 +621,6 @@ export has_transitive_groups
 export has_vertex
 export haspreimage
 export height
-export helper_eta
-export helper_ideal
-export helper_images
-export helper_kappa
-export helper_ring
 export hermitian_form
 export hessian
 export hessian_matrix
@@ -685,7 +661,6 @@ export ideal_as_module
 export ideal_membership
 export ideal_of_linear_relations
 export ideal_sheaf
-export ideal_type
 export identifier
 export identity_map
 export image
@@ -1005,7 +980,6 @@ export min_revlex_basis_encoding
 export min_weights
 export minimal_betti_table
 export minimal_block_reps
-export minimal_denominators
 export minimal_faces
 export minimal_generating_set, has_minimal_generating_set, set_minimal_generating_set
 export minimal_generators
@@ -1040,7 +1014,6 @@ export mpoly_dec_type
 export mpoly_ring_type
 export mpoly_type
 export mul!
-export mult_set_type
 export multi_hilbert_function
 export multi_hilbert_series
 export multi_hilbert_series_reduced
@@ -1316,6 +1289,7 @@ export right_coset_action
 export right_cosets
 export right_ideal
 export right_transversal
+export ring
 export ring_elem_type
 export ring_type
 export rising_factorial
@@ -1523,6 +1497,5 @@ export weyl_vector
 export witt_index
 export wreath_product
 export write_as_full
-export write_as_linear_combination
 export zonotope
 export zonotope_vertices_fukuda_matrix
diff --git a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
index c892b3cfe3c6..d06ba04094ae 100644
--- a/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/AffineSchemes.jl
@@ -194,9 +194,9 @@ end
   @test_broken issubset(intersect(A3,V),A3)
   @test ambient_coordinate_ring(V)===R
   @test ambient_coordinate_ring(U) === R
-  @test ring_type(V) == typeof(OO(V))
-  @test base_ring_type(typeof(V)) == typeof(QQ)
-  @test base_ring_elem_type(V) == QQFieldElem
+  @test Oscar.ring_type(V) == typeof(OO(V))
+  @test Oscar.base_ring_type(typeof(V)) == typeof(QQ)
+  @test Oscar.base_ring_elem_type(V) == QQFieldElem
   @test base_ring(V) == QQ
   @test issubset(V,U)
   @test issubset(U,V)
diff --git a/test/AlgebraicGeometry/Schemes/SpaceGerms.jl b/test/AlgebraicGeometry/Schemes/SpaceGerms.jl
index e8da7ec267ff..87e4a19f9ca8 100644
--- a/test/AlgebraicGeometry/Schemes/SpaceGerms.jl
+++ b/test/AlgebraicGeometry/Schemes/SpaceGerms.jl
@@ -12,8 +12,8 @@
   @test defining_ring_element(Z0) isa elem_type(Oscar.localized_ring_type(ring_type(Z0)))
   Z1 = CompleteIntersectionGerm(X,[1,2,1])
   @test defining_ring_elements(Z1) isa Vector{elem_type(Oscar.localized_ring_type(ring_type(Z0)))}
-  U0 = MPolyComplementOfKPointIdeal(R,[0,0,0])
-  U1 = MPolyComplementOfKPointIdeal(R,[1,2,1])
+  U0 = Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
+  U1 = Oscar.MPolyComplementOfKPointIdeal(R,[1,2,1])
   @test U0 == inverted_set(OO(X0))
   @test U1 == inverted_set(OO(X1))
   @test X0 == Y0
@@ -55,8 +55,8 @@ end
 @testset "Space Germ constructors Spec-Ideal" begin
   R, (x,y,z) = QQ["x", "y", "z"]
   J = ideal(R, [x-y])
-  U0 = MPolyComplementOfKPointIdeal(R,[0,0,0])
-  U1 = MPolyComplementOfKPointIdeal(R,[1,2,2])
+  U0 = Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
+  U1 = Oscar.MPolyComplementOfKPointIdeal(R,[1,2,2])
   Xg = Spec(R)
   Xgq = Spec(R,J)
   Xl1 = Spec(R, U0)
@@ -80,8 +80,8 @@ end
   X1 = SpaceGerm(X,ideal(R,[x-1,y-2,z-2]))
   Y0 = SpaceGerm(Y,ideal(R,[x,y,z]))
   Y1 = SpaceGerm(Y,ideal(R,[x-1,y-2,z-2]))
-  U0 = MPolyComplementOfKPointIdeal(R,[0,0,0])
-  U1 = MPolyComplementOfKPointIdeal(R,[1,2,2])
+  U0 = Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
+  U1 = Oscar.MPolyComplementOfKPointIdeal(R,[1,2,2])
   @test U0 == inverted_set(OO(X0))
   @test U1 == inverted_set(OO(X1))
   @test X0 == Y0
@@ -118,7 +118,7 @@ end
   X0,phi0 = germ_at_point(X,ideal(R,[x,y,z]))
   X1 = SpaceGerm(X,ideal(R,[x,y,z]))
   X2,phi2 = germ_at_point(X,[0,0,0])
-  U0 = MPolyComplementOfKPointIdeal(R,[0,0,0])
+  U0 = Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
   @test U0 == inverted_set(OO(X0))
   @test U0 == inverted_set(OO(X2))
   @test X0 == X1
@@ -161,7 +161,7 @@ end
   @test defining_ideal(X0) != ideal(R,zero(R))
   @test defining_ideal(X0) == ideal(OO(X0),[0])
   @test defining_ideal(X2) == ideal(localized_ring(OO(X2)),[x,y])
-  @test inverted_set(OO(ambient_germ(X2))) == MPolyComplementOfKPointIdeal(R,[0,0,0])
+  @test inverted_set(OO(ambient_germ(X2))) == Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
   @test ambient_germ(X0) == X0
 end
 
diff --git a/test/Modules/ProjectiveModules.jl b/test/Modules/ProjectiveModules.jl
index d5daae38aa93..ea69a6bfe81f 100644
--- a/test/Modules/ProjectiveModules.jl
+++ b/test/Modules/ProjectiveModules.jl
@@ -13,7 +13,7 @@
   @test success
   @test P^2 == P
 
-  W = MPolyQuoLocRing(R, I, units_of(R))
+  W = Oscar.MPolyQuoLocRing(R, I, units_of(R))
   F3 = FreeMod(W, 3)
   F4 = FreeMod(W, 4)
   h = hom(F3, F4, map_entries(W, M))
diff --git a/test/Modules/UngradedModules.jl b/test/Modules/UngradedModules.jl
index 4d9732a0b90f..f605b2fef066 100644
--- a/test/Modules/UngradedModules.jl
+++ b/test/Modules/UngradedModules.jl
@@ -1019,8 +1019,8 @@ end
 
 @testset "change of base rings" begin
   R, (x,y) = QQ["x", "y"]
-  U = MPolyPowersOfElement(x)
-  S = MPolyLocRing(R, U)
+  U = Oscar.MPolyPowersOfElement(x)
+  S = Oscar.MPolyLocRing(R, U)
   F = FreeMod(R, 2)
   FS, mapF = change_base_ring(S, F)
   @test 1//x*mapF(x*F[1]) == FS[1]
diff --git a/test/Modules/module-localizations.jl b/test/Modules/module-localizations.jl
index a179a2ca4558..e2f28ccfc996 100644
--- a/test/Modules/module-localizations.jl
+++ b/test/Modules/module-localizations.jl
@@ -1,7 +1,7 @@
 @testset "module localizations 1" begin
   kk = QQ
   R, (x,y) = QQ["x", "y"]
-  U = MPolyComplementOfKPointIdeal(R, [0, 0])
+  U = Oscar.MPolyComplementOfKPointIdeal(R, [0, 0])
   L, _ = localization(U)
   F = FreeMod(L, 3)
   A = L[x 0 1; 0 y y^2]
@@ -17,7 +17,7 @@
   @test represents_element((x^8-9)*F[2], K)
   @test represents_element(x*F[3], K)
 
-  T = MPolyPowersOfElement(R, [x, y])
+  T = Oscar.MPolyPowersOfElement(R, [x, y])
   W, _ = localization(T)
   F = FreeMod(W, 2)
   A = W[x 0; 0 y^2]
@@ -28,10 +28,10 @@ end
 
 @testset "module localizations 2" begin
   R, (x,y) = QQ["x", "y"]
-  U = MPolyPowersOfElement(x+y)
+  U = Oscar.MPolyPowersOfElement(x+y)
   S, _ = localization(U)
   F = FreeMod(S, 2)
-  Fb = base_ring_module(F)
+  Fb = Oscar.base_ring_module(F)
   A = S[x//(x+y); y//(x+y)^2]
   B, D = Oscar.clear_denominators(A)
   @test change_base_ring(S, D) * A == B
@@ -41,7 +41,7 @@ end
   @test success
   @test v*A == b
 
-  V = MPolyComplementOfPrimeIdeal(ideal(R, [x,y]))
+  V = Oscar.MPolyComplementOfPrimeIdeal(ideal(R, [x,y]))
   S, _ = localization(V)
   A = S[x//(x+y+1); y*(x-5)^3]
   b = matrix(S, 1, 1, [(x+y)*x + 5*y//(x+y+2)^10])
@@ -75,13 +75,13 @@ end
 
 @testset "module localizations 3" begin
   R, (x,y) = QQ["x", "y"]
-  U = MPolyPowersOfElement(x^7)
+  U = Oscar.MPolyPowersOfElement(x^7)
   S, _ = localization(U)
   F = FreeMod(S, 1)
   A = S[x^4*y^2; x^2*y]
   B = S[y^8; y^9]
   M = SubquoModule(F, A, B)
-  Fb = base_ring_module(F)
+  Fb = Oscar.base_ring_module(F)
   @test !represents_element(y*Fb[1], Oscar.pre_saturated_module(M))
   @test represents_element(y*F[1], M)
   v = coordinates(y*F[1], M)
@@ -107,7 +107,7 @@ end
   R, (x,y,z) = QQ["x", "y", "z"]
   M = R[x-1 y; z x]
   I = ideal(R, det(M))
-  Q = MPolyQuoLocRing(R, I, units_of(R))
+  Q = Oscar.MPolyQuoLocRing(R, I, units_of(R))
   A = map_entries(Q, M)
   K1 = syz(A)
   K2 = syz(K1)
diff --git a/test/Rings/MPolyQuo.jl b/test/Rings/MPolyQuo.jl
index 8e384bd858fb..9e2f0c994bc8 100644
--- a/test/Rings/MPolyQuo.jl
+++ b/test/Rings/MPolyQuo.jl
@@ -168,32 +168,6 @@ end
   @test !is_prime(J2)
 end
 
-@testset "modulus - MPAnyQuoRing MPAnyNonQuoRing" begin
-  R, (x,y,z) = QQ["x", "y", "z"]
-  I = ideal(R, [x+y+z])
-  A, _ = quo(R,I)
-  @test modulus(R) == ideal(R,[zero(R)])
-  @test modulus(A) == I
-  U=MPolyComplementOfKPointIdeal(R,[0,0,0])
-  Rl,_ = localization(R,U)
-  Il = Rl(I)
-  Al, _ = quo(Rl, Il)
-  @test modulus(Rl) == ideal(Rl,[zero(Rl)])
-  @test modulus(Al) == Il
-  U2=MPolyComplementOfPrimeIdeal(ideal(R,[x^2+1,y-x,z]))
-  Rl2,_ = localization(R,U2)
-  Il2 = Rl2(I)
-  Al2,_ = quo(Rl2,Il2)
-  @test modulus(Rl2) == ideal(Rl2,[zero(Rl2)])
-  @test modulus(Al2) == Il2
-  U3=MPolyPowersOfElement(x+y)
-  Rl3,_ = localization(R,U3)
-  Il3 = Rl3(I)
-  Al3,_ = quo(Rl3,Il3)
-  @test modulus(Rl3) == ideal(Rl3,[zero(Rl3)])
-  @test modulus(Al3) == Il3
-end
-
 @testset "saturated ideal compatibility" begin
   R, (x,y,z) = QQ["x", "y", "z"]
   I = ideal(R, [x+y+z])
diff --git a/test/Rings/integer-localizations.jl b/test/Rings/integer-localizations.jl
index 037a0a40c01c..212982071504 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, ambient_ring, localization, parent, numerator, denominator, one, zero, reduce_fraction
+import Oscar: base_ring, inverted_set, ring, localization, parent, numerator, denominator, one, zero, reduce_fraction
 import Oscar.AbstractAlgebra: elem_type, parent_type
 
 export FmpzComplementOfPrimeIdeal, FmpzPowersOfElement, FmpzComplementOfZeroIdeal
@@ -29,7 +29,7 @@ mutable struct FmpzPowersOfElement <: AbsMultSet{ZZRing, ZZRingElem}
 end
 
 ### required getter functions
-ambient_ring(S::FmpzPowersOfElement) = ZZ
+ring(S::FmpzPowersOfElement) = ZZ
 
 ### required functionality
 function Base.in(a::ZZRingElem, S::FmpzPowersOfElement) 
@@ -73,7 +73,7 @@ mutable struct FmpzComplementOfPrimeIdeal <: AbsMultSet{ZZRing, ZZRingElem}
 end
 
 ### required getter functions
-ambient_ring(S::FmpzComplementOfPrimeIdeal) = ZZ
+ring(S::FmpzComplementOfPrimeIdeal) = ZZ
 
 ### required functionality
 function Base.in(b::ZZRingElem, S::FmpzComplementOfPrimeIdeal)
@@ -103,7 +103,7 @@ mutable struct FmpzComplementOfZeroIdeal <: AbsMultSet{ZZRing, ZZRingElem}
 end
 
 ### required getter functions
-ambient_ring(S::FmpzComplementOfZeroIdeal) = ZZ
+ring(S::FmpzComplementOfZeroIdeal) = ZZ
 
 ### required functionality
 function Base.in(b::ZZRingElem, S::FmpzComplementOfZeroIdeal) 
@@ -240,9 +240,9 @@ parent_type(T::Type{FmpzLocalizedRingElem{MultSetType}}) where {MultSetType} = F
   @test 25 in S
   @test !(33 in S)
   @test 5*5*7 in S
-  @test ambient_ring(S) == ZZ
+  @test ring(S) == ZZ
   W, _ = localization(S)
-  @test base_ring(W) == ambient_ring(S)
+  @test base_ring(W) == ring(S)
   @test inverted_set(W) == S
   a = W(3)
   b = W(7, 5)
@@ -257,9 +257,9 @@ parent_type(T::Type{FmpzLocalizedRingElem{MultSetType}}) where {MultSetType} = F
   @test !(13^5 in U)
   @test !(13*4289729837 in U)
   @test 5783790198374098 in U
-  @test ambient_ring(U) == ZZ
+  @test ring(U) == ZZ
   W, _ = localization(U)
-  @test base_ring(W) == ambient_ring(U)
+  @test base_ring(W) == ring(U)
   @test inverted_set(W) == U
   a = W(4, 17)
   b = W(4*17)
@@ -272,9 +272,9 @@ parent_type(T::Type{FmpzLocalizedRingElem{MultSetType}}) where {MultSetType} = F
   O = FmpzComplementOfZeroIdeal()
   @test 234890 in O
   @test !(0 in O)
-  @test ambient_ring(O) == ZZ
+  @test ring(O) == ZZ
   W, _ = localization(O)
-  @test base_ring(W) == ambient_ring(O)
+  @test base_ring(W) == ring(O)
   @test inverted_set(W) == O
   a = W(4, 17)
   b = W(4*17)
diff --git a/test/Rings/mpoly-localizations.jl b/test/Rings/mpoly-localizations.jl
index 7036345a93f7..31e3d81f95a5 100644
--- a/test/Rings/mpoly-localizations.jl
+++ b/test/Rings/mpoly-localizations.jl
@@ -7,9 +7,9 @@ const rng = Oscar.get_seeded_rng()
   f = x^2 + y^2 -1
   m = ideal(R, [x, y])
   I = ideal(R, f)
-  S = MPolyComplementOfPrimeIdeal(I)
+  S = Oscar.MPolyComplementOfPrimeIdeal(I)
   V, _ = localization(S)
-  T = MPolyComplementOfKPointIdeal(R, [ZZ(1), ZZ(0)])
+  T = Oscar.MPolyComplementOfKPointIdeal(R, [ZZ(1), ZZ(0)])
   W, _ = localization(T)
   
   k = QQ
@@ -21,16 +21,16 @@ const rng = Oscar.get_seeded_rng()
   f = x^2 + y^2 - p^2 - q^2
   m = ideal(R, [x, y])
   I = ideal(R, f)
-  S = MPolyComplementOfPrimeIdeal(I)
-  @test ambient_ring(S) == R
+  S = Oscar.MPolyComplementOfPrimeIdeal(I)
+  @test ring(S) == R
   @test !( f in S )
   @test x in S
   V, _ = localization(S)
   @test base_ring(V) == R
   @test inverted_set(V) == S
 
-  T = MPolyComplementOfKPointIdeal(R, [k(p), k(q)])
-  @test ambient_ring(T) == R
+  T = Oscar.MPolyComplementOfKPointIdeal(R, [k(p), k(q)])
+  @test ring(T) == R
   @test typeof(point_coordinates(T)) == Vector{elem_type(k)}
   @test x in T
   @test !(x-p in T)
@@ -55,8 +55,8 @@ const rng = Oscar.get_seeded_rng()
   J = ideal(W, [W(x-3), W(y-4)])
 
   J = ideal(W, [f, y-q])
-  @test I_loc + J isa MPolyLocalizedIdeal
-  @test I_loc * J isa MPolyLocalizedIdeal
+  @test I_loc + J isa Oscar.MPolyLocalizedIdeal
+  @test I_loc * J isa Oscar.MPolyLocalizedIdeal
 
  # @test reduce(W(x)//W(y-q+1), lbpa) == W(p)//W(y-q+1)
 
@@ -72,9 +72,9 @@ const rng = Oscar.get_seeded_rng()
   x = v[1]
   y = v[2] 
   f = (x^2 + y^2)^2
-  S = MPolyComplementOfKPointIdeal(R, [ZZ(0), ZZ(0)])
-  T = MPolyPowersOfElement(R, [f])
-  U = MPolyProductOfMultSets(R, [S, T])
+  S = Oscar.MPolyComplementOfKPointIdeal(R, [ZZ(0), ZZ(0)])
+  T = Oscar.MPolyPowersOfElement(R, [f])
+  U = Oscar.MPolyProductOfMultSets(R, [S, T])
   @test f in U
   @test (f*(x-1) in U)
   @test !(f*x in U)
@@ -87,8 +87,8 @@ const rng = Oscar.get_seeded_rng()
   I = ideal(S, [x + y + z, x^2 + y^2 + z^3])
 
   R, (x, y) = QQ["x", "y"]
-  U = MPolyPowersOfElement(R, [x])
-  L = MPolyLocRing(R, U)
+  U = Oscar.MPolyPowersOfElement(R, [x])
+  L = Oscar.MPolyLocRing(R, U)
   I = ideal(L, [y*(y-x)])
   J = ideal(L, [x*(y-x)])
   @test y in quotient(I, J)
@@ -97,8 +97,8 @@ const rng = Oscar.get_seeded_rng()
   R, (x, y) = QQ["x", "y"]
   f = x^2 + y^3- 2
   I = ideal(R, [f])
-  U = MPolyComplementOfPrimeIdeal(I)
-  L = MPolyLocRing(R, U)
+  U = Oscar.MPolyComplementOfPrimeIdeal(I)
+  L = Oscar.MPolyLocRing(R, U)
   I = ideal(L, [f^4])
   J = ideal(L, [f^2])
   @test f^2 in quotient(I, J)
@@ -108,8 +108,8 @@ const rng = Oscar.get_seeded_rng()
   R, (x, y) = QQ["x", "y"]
   f = x^2 + y^2- 2
   I = ideal(R, [f])
-  U = MPolyComplementOfKPointIdeal(R, [1, 1])
-  L = MPolyLocRing(R, U)
+  U = Oscar.MPolyComplementOfKPointIdeal(R, [1, 1])
+  L = Oscar.MPolyLocRing(R, U)
   I = ideal(L, [y^2*f^4])
   J = ideal(L, [x*f^2])
   @test f^2 in quotient(I, J)
@@ -122,7 +122,7 @@ end
   x = variab[1]
   y = variab[2] 
   f = x^2 + y^2 -1
-  S = MPolyPowersOfElement(R, [x, y, f])
+  S = Oscar.MPolyPowersOfElement(R, [x, y, f])
   @test f in S
   # 5 is not a unit in R
   @test !(5*f in S)
@@ -131,7 +131,7 @@ end
   x = variabs[1]
   y = variabs[2]
   f = x^2 + y^4-120
-  S = MPolyPowersOfElement(R, [x, y, f])
+  S = Oscar.MPolyPowersOfElement(R, [x, y, f])
   @test f in S
   # Now 5 is a unit in R
   @test (5*f in S)
@@ -154,7 +154,7 @@ end
   x = variab[1]
   y = variab[2] 
   f = x^2 + y^2 -1
-  S = MPolyPowersOfElement(R, [x, y, f])
+  S = Oscar.MPolyPowersOfElement(R, [x, y, f])
   @test f in S
   @test !(5*f in S)
 
@@ -162,15 +162,15 @@ end
   V, _ = localization(U, x)
   W, _ = localization(V, 5*f)
 
-  phi = MPolyLocalizedRingHom(R, W, [W(1//x), W(y//f)])
+  phi = Oscar.MPolyLocalizedRingHom(R, W, [W(1//x), W(y//f)])
   @test phi(x*f) == phi(domain(phi)(x*f))
   @test phi(ZZ(9)) == W(ZZ(9))
 
-  psi = MPolyLocalizedRingHom(U, R, [R(0), R(0)])
+  psi = Oscar.MPolyLocalizedRingHom(U, R, [R(0), R(0)])
   @test psi(U(-1//f))==one(codomain(psi))
 end
 
-function test_elem(W::MPolyLocRing) 
+function test_elem(W::Oscar.MPolyLocRing) 
   f = rand(rng, W, 0:3, 0:4, 0:3)
   return f
 end
@@ -185,9 +185,9 @@ end
 #
 # d = Vector{elem_type(R)}()
 # d = [rand(R, 1:3, 0:4, 1:10)::elem_type(R) for i in 0:(abs(rand(Int))%3+1)]
-# S = MPolyPowersOfElement(R, d)
-# T = MPolyComplementOfKPointIdeal(R, [kk(125), kk(-45)])
-# U = MPolyComplementOfPrimeIdeal(I)
+# S = Oscar.MPolyPowersOfElement(R, d)
+# T = Oscar.MPolyComplementOfKPointIdeal(R, [kk(125), kk(-45)])
+# U = Oscar.MPolyComplementOfPrimeIdeal(I)
 #
 # test_Ring_interface_recursive(localization(S)[1])
 # test_Ring_interface_recursive(localization(T)[1])
@@ -209,9 +209,9 @@ end
     f = rand(rng, R, 1:3, 0:3, 1:10)::elem_type(R)
     iszero(f) || push!(d, f)
   end
-  S = MPolyPowersOfElement(R, d)
-  T = MPolyComplementOfKPointIdeal(R, [kk(125), kk(-45)])
-  U = MPolyComplementOfPrimeIdeal(I)
+  S = Oscar.MPolyPowersOfElement(R, d)
+  T = Oscar.MPolyComplementOfKPointIdeal(R, [kk(125), kk(-45)])
+  U = Oscar.MPolyComplementOfPrimeIdeal(I)
 
   test_Ring_interface_recursive(localization(S)[1])
   test_Ring_interface_recursive(localization(T)[1])
@@ -226,9 +226,9 @@ end
 #
 # d = Vector{elem_type(R)}()
 # d = [rand(R, 1:3, 0:4, 1:10)::elem_type(R) for i in 0:(abs(rand(Int))%3+1)]
-# S = MPolyPowersOfElement(R, d)
-# T = MPolyComplementOfKPointIdeal(R, [kk(125), kk(-45)])
-# U = MPolyComplementOfPrimeIdeal(I)
+# S = Oscar.MPolyPowersOfElement(R, d)
+# T = Oscar.MPolyComplementOfKPointIdeal(R, [kk(125), kk(-45)])
+# U = Oscar.MPolyComplementOfPrimeIdeal(I)
 #
 # test_Ring_interface_recursive(localization(S)[1])
 # test_Ring_interface_recursive(localization(T)[1])
@@ -238,7 +238,7 @@ end
 @testset "localization_at_orderings_1" begin
   R, (x,y) = QQ["x", "y"]
   o = degrevlex([x])*negdegrevlex([y])
-  U = MPolyLeadingMonOne(R, o)
+  U = Oscar.MPolyLeadingMonOne(R, o)
   @test y-1 in U
   @test !(x in U)
   L, _ = localization(R, U)
@@ -264,7 +264,7 @@ end
 @testset "localizations at k-points" begin
   R, (x, y, z) = QQ["x", "y", "z"]
   p = [-5, 8, 1//2]
-  U = MPolyComplementOfKPointIdeal(R, p)
+  U = Oscar.MPolyComplementOfKPointIdeal(R, p)
   I = ideal(R, [x*(x+5), (y-8)*y-z*(x+5)])
   L, _ = localization(R, U)
   LI = L(I)
@@ -283,19 +283,19 @@ end
 @testset "successive localizations" begin
   R, (x, y, z) = QQ["x", "y", "z"]
   p = [0,0,0]
-  U = MPolyComplementOfKPointIdeal(R, p)
+  U = Oscar.MPolyComplementOfKPointIdeal(R, p)
   I = ideal(R, [x*(y-1)-z*(x-2), y*x])
   L, _ = localization(R, U)
   LI = L(I)
   W, _ = quo(L, LI)
-  S = MPolyPowersOfElement(R, [y])
+  S = Oscar.MPolyPowersOfElement(R, [y])
   RS, _ = localization(R, S)
   RSI = RS(I)
   saturated_ideal(RSI, with_generator_transition=true)
   J = L(Oscar.pre_saturated_ideal(RSI))
   z in J
   W, _ = quo(L, LI)
-  S = MPolyPowersOfElement(R, [y])
+  S = Oscar.MPolyPowersOfElement(R, [y])
   WS, _ = localization(W, S)
   @test !iszero(W(z))
   @test iszero(WS(z))
@@ -310,8 +310,8 @@ end
 @testset "saturated_ideals" begin
   R, (x, y) = QQ["x", "y"]
   I = ideal(R, [x, y^2+1])
-  U = MPolyComplementOfPrimeIdeal(I)
-  L = MPolyLocRing(R, U)
+  U = Oscar.MPolyComplementOfPrimeIdeal(I)
+  L = Oscar.MPolyLocRing(R, U)
   J = ideal(L,[y*(x^2+(y^2+1)^2)])
   J_sat = ideal(R,[(x^2+(y^2+1)^2)])
   @test saturated_ideal(J) == J_sat
diff --git a/test/Rings/mpolyquo-localizations.jl b/test/Rings/mpolyquo-localizations.jl
index 85420d741710..c6c660a91162 100644
--- a/test/Rings/mpolyquo-localizations.jl
+++ b/test/Rings/mpolyquo-localizations.jl
@@ -7,8 +7,8 @@
   f = x*v-y*u
   I = ideal(R, f)
   Q, p = quo(R, I)
-  S = MPolyComplementOfKPointIdeal(R, [QQ(1), QQ(0), QQ(1), QQ(0)])
-  T = MPolyComplementOfKPointIdeal(R, [QQ(0), QQ(0), QQ(0), QQ(0)])
+  S = Oscar.MPolyComplementOfKPointIdeal(R, [QQ(1), QQ(0), QQ(1), QQ(0)])
+  T = Oscar.MPolyComplementOfKPointIdeal(R, [QQ(0), QQ(0), QQ(0), QQ(0)])
   L, _ = localization(Q, S)
   a = L(x)
   b = L(y)
@@ -20,16 +20,16 @@
   x = v[1]
   y = v[2] 
   f = (x^2 + y^2)
-  T = MPolyComplementOfKPointIdeal(R, [kk(0), kk(0)])
+  T = Oscar.MPolyComplementOfKPointIdeal(R, [kk(0), kk(0)])
   I = ideal(R, f)
-  V = MPolyQuoLocRing(R, I, T)
+  V = Oscar.MPolyQuoLocRing(R, I, T)
 
   S = R
-  U = MPolyPowersOfElement(S, [f-1])
+  U = Oscar.MPolyPowersOfElement(S, [f-1])
   J = ideal(S, zero(S))
-  W = MPolyQuoLocRing(S, J, U)
+  W = Oscar.MPolyQuoLocRing(S, J, U)
 
-  h = MPolyQuoLocalizedRingHom(W, V, [x//(y-1), y//(x-5)])
+  h = Oscar.MPolyQuoLocalizedRingHom(W, V, [x//(y-1), y//(x-5)])
   J1 = ideal(V, [x*(x-1), y*(y-3)])
   J2 = ideal(W, [x, y])
   @test preimage(h, J1) == J2
@@ -41,20 +41,20 @@
   R, (x,y) = kk["x", "y"]
 
   f = x^2 + y^2-2
-  S = MPolyPowersOfElement(x-1)
-  T = MPolyComplementOfKPointIdeal(R, [1,1])
-  V = MPolyComplementOfPrimeIdeal(ideal(R, f))
+  S = Oscar.MPolyPowersOfElement(x-1)
+  T = Oscar.MPolyComplementOfKPointIdeal(R, [1,1])
+  V = Oscar.MPolyComplementOfPrimeIdeal(ideal(R, f))
   ⊂ = issubset
   @test S ⊂ V
   @test !(V ⊂ S)
   @test !(T ⊂ V)
   @test (V ⊂ T)
-  @test !(MPolyComplementOfPrimeIdeal(ideal(R, f-1)) ⊂ T)
-  @test S ⊂ MPolyComplementOfPrimeIdeal(ideal(R, f-1))
-  @test !(MPolyPowersOfElement(f) ⊂ V)
-  @test MPolyPowersOfElement(x-1) ⊂ MPolyComplementOfKPointIdeal(R, [0,0])
-  @test MPolyPowersOfElement(x-1) * MPolyComplementOfKPointIdeal(R, [0,0]) ⊂ MPolyComplementOfKPointIdeal(R, [0,0])
-  @test !(MPolyPowersOfElement(x) ⊂ MPolyComplementOfKPointIdeal(R, [0,0]))
+  @test !(Oscar.MPolyComplementOfPrimeIdeal(ideal(R, f-1)) ⊂ T)
+  @test S ⊂ Oscar.MPolyComplementOfPrimeIdeal(ideal(R, f-1))
+  @test !(Oscar.MPolyPowersOfElement(f) ⊂ V)
+  @test Oscar.MPolyPowersOfElement(x-1) ⊂ Oscar.MPolyComplementOfKPointIdeal(R, [0,0])
+  @test Oscar.MPolyPowersOfElement(x-1) * Oscar.MPolyComplementOfKPointIdeal(R, [0,0]) ⊂ Oscar.MPolyComplementOfKPointIdeal(R, [0,0])
+  @test !(Oscar.MPolyPowersOfElement(x) ⊂ Oscar.MPolyComplementOfKPointIdeal(R, [0,0]))
   @test T*T == T
 
   U = S*T
@@ -83,7 +83,7 @@
 
   h = x^4+23*x*y^3-15
   Q, _ = quo(R, f)
-  T = MPolyPowersOfElement(h^3)
+  T = Oscar.MPolyPowersOfElement(h^3)
   W, _ = localization(Q, T)
   @test x//(h+3*f) in W
   @test W(x//(h+3*f)) == W(x//h)
@@ -92,21 +92,21 @@
   g = [W(8*x*h+4), W(x, 45*h^2)]
   @test one(localized_ring(W)) in ideal(W, g)
 
-  @test one(W) == dot(write_as_linear_combination(one(W), g), g)
+  @test one(W) == dot(Oscar.write_as_linear_combination(one(W), g), g)
 
 
   h = (x+5)*(x^2+10*y)+(y-7)*(y^2-3*x)
   Q, _ = quo(R, h)
-  T = MPolyComplementOfKPointIdeal(R, [-5, 7])
+  T = Oscar.MPolyComplementOfKPointIdeal(R, [-5, 7])
   W, _ = localization(Q, T)
   @test x//(y) in W
   @test x//(y+h) in W
   g = [W(h + (x+5) - 9, y+24*x^3-8)]
-  (d, a) = bring_to_common_denominator([one(W), g[1]])
+  (d, a) = Oscar.bring_to_common_denominator([one(W), g[1]])
   @test W(a[1], d) == one(W)
   @test W(a[2]*lifted_numerator(g[1]), d) == g[1]
   @test (one(localized_ring(W)) in ideal(W, g))
-  @test one(W) == dot(write_as_linear_combination(one(W), g), g) 
+  @test one(W) == dot(Oscar.write_as_linear_combination(one(W), g), g) 
 end
 
 @testset "prime ideals in quotient rings" begin
@@ -148,9 +148,9 @@ end
   Q2 = ideal(R,[x*y-z*w])
   RQ1,phiQ1 = quo(R,Q1)
   RQ2,phiQ2 = quo(R,Q2)
-  T1 = MPolyComplementOfKPointIdeal(R,[0,0,0,0])
+  T1 = Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0,0])
   f = x+y+z+w-1
-  T2 = MPolyPowersOfElement(f)
+  T2 = Oscar.MPolyPowersOfElement(f)
   RL1,phiL1 = localization(R,T1)
   RL2,phiL2 = localization(R,T2)
   RQ1L1, phiQ1L1 = localization(RQ1,T1)
@@ -206,9 +206,9 @@ end
   Q2 = ideal(R,[z,x^2-y^2])
   RQ1,phiQ1 = quo(R,Q1)
   RQ2,phiQ2 = quo(R,Q2)
-  T1 = MPolyComplementOfKPointIdeal(R,[0,0,0])
+  T1 = Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
   f = x-y
-  T2 = MPolyPowersOfElement(f)
+  T2 = Oscar.MPolyPowersOfElement(f)
   RL1,phiL1 = localization(R,T1)
   RL2,phiL2 = localization(R,T2)
   RQ1L1, phiQ1L1 = localization(RQ1,T1)
@@ -299,12 +299,12 @@ end
 @testset "minimal and small generating sets" begin
   R, (x,y,z) = QQ["x","y","z"]
   IQ = ideal(R,[x-z])
-  U1 = MPolyComplementOfKPointIdeal(R,[0,0,0])
-  U2 = MPolyComplementOfKPointIdeal(R,[1,1,1])
-  U3 = MPolyPowersOfElement(y)
-  Q1 = MPolyQuoLocRing(R,IQ,U1)
-  Q2 = MPolyQuoLocRing(R,IQ,U2)
-  Q3 = MPolyQuoLocRing(R,IQ,U3)
+  U1 = Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
+  U2 = Oscar.MPolyComplementOfKPointIdeal(R,[1,1,1])
+  U3 = Oscar.MPolyPowersOfElement(y)
+  Q1 = Oscar.MPolyQuoLocRing(R,IQ,U1)
+  Q2 = Oscar.MPolyQuoLocRing(R,IQ,U2)
+  Q3 = Oscar.MPolyQuoLocRing(R,IQ,U3)
   J1 = ideal(Q1,[x^2-y^2,y^2-z^2,x^2-z^2])
   @test length(minimal_generating_set(J1)) == 1
   @test length(small_generating_set(J1)) == 1
@@ -321,7 +321,7 @@ end
   R, (x, y) = QQ[:x, :y]
   I = ideal(R, x^6-y)
   U = complement_of_point_ideal(R, [1, 1])
-  L = MPolyQuoLocRing(R, I, U)
+  L = Oscar.MPolyQuoLocRing(R, I, U)
   J = ideal(L, [x-1, y-1])^2
   minJ = minimal_generating_set(J)
   @test length(minJ) == 1
@@ -476,3 +476,30 @@ end
   L_to_Tl3 = compose(L_to_Sl3, Sl_to_Tl3)
   @test L_to_Tl(one(L)) == one(Tl)
 end
+
+@testset "modulus - MPAnyQuoRing MPAnyNonQuoRing" begin
+  R, (x,y,z) = QQ["x", "y", "z"]
+  I = ideal(R, [x+y+z])
+  A, _ = quo(R,I)
+  @test modulus(R) == ideal(R,[zero(R)])
+  @test modulus(A) == I
+  U= Oscar.MPolyComplementOfKPointIdeal(R,[0,0,0])
+  Rl,_ = localization(R,U)
+  Il = Rl(I)
+  Al, _ = quo(Rl, Il)
+  @test modulus(Rl) == ideal(Rl,[zero(Rl)])
+  @test modulus(Al) == Il
+  U2= Oscar.MPolyComplementOfPrimeIdeal(ideal(R,[x^2+1,y-x,z]))
+  Rl2,_ = localization(R,U2)
+  Il2 = Rl2(I)
+  Al2,_ = quo(Rl2,Il2)
+  @test modulus(Rl2) == ideal(Rl2,[zero(Rl2)])
+  @test modulus(Al2) == Il2
+  U3= Oscar.MPolyPowersOfElement(x+y)
+  Rl3,_ = localization(R,U3)
+  Il3 = Rl3(I)
+  Al3,_ = quo(Rl3,Il3)
+  @test modulus(Rl3) == ideal(Rl3,[zero(Rl3)])
+  @test modulus(Al3) == Il3
+end
+
diff --git a/test/Rings/nmod-localizations.jl b/test/Rings/nmod-localizations.jl
index 4c4f06df56d3..f677ec2c8164 100644
--- a/test/Rings/nmod-localizations.jl
+++ b/test/Rings/nmod-localizations.jl
@@ -5,12 +5,12 @@
 using Oscar
 
 import Oscar.Nemo.zzModRing
-import Oscar: base_ring, inverted_set, ambient_ring, localization, parent, numerator, denominator, one, zero
+import Oscar: base_ring, inverted_set, ring, localization, parent, numerator, denominator, one, zero
 import Oscar.AbstractAlgebra: elem_type, parent_type
 
 export NmodComplementOfPrimeIdeal, NmodLocalizedRing, NmodLocalizedRingElem
 
-export generator, ambient_ring, localization, parent, numerator, denominator
+export generator, ring, localization, parent, numerator, denominator
 
 
 #######################################################################
@@ -42,14 +42,14 @@ function Base.in(b::zzModRingElem, S::NmodComplementOfPrimeIdeal)
 end
 
 ### required getter functions
-ambient_ring(S::NmodComplementOfPrimeIdeal) = S.R
+ring(S::NmodComplementOfPrimeIdeal) = S.R
 
 ### additional constructors
 NmodComplementOfPrimeIdeal(R::zzModRing, i::Oscar.IntegerUnion) = NmodComplementOfPrimeIdeal(R(i))
 
 ### additional functionality
 generator(S::NmodComplementOfPrimeIdeal) = S.gen
-Base.in(b::Oscar.IntegerUnion, S::NmodComplementOfPrimeIdeal) = (ambient_ring(S)(b) in S)
+Base.in(b::Oscar.IntegerUnion, S::NmodComplementOfPrimeIdeal) = (ring(S)(b) in S)
 
 #######################################################################
 # Localizations of ℤ/nℤ                                               #
@@ -65,7 +65,7 @@ mutable struct NmodLocalizedRing{MultSetType <: AbsMultSet{zzModRing, zzModRingE
   S::MultSetType # the set at which has been localized
 
   function NmodLocalizedRing(S::MultSetType) where {MultSetType <: AbsMultSet{zzModRing, zzModRingElem}}
-    return new{MultSetType}(ambient_ring(S), S)
+    return new{MultSetType}(ring(S), S)
   end
 end
 
@@ -151,9 +151,9 @@ parent_type(T::Type{NmodLocalizedRingElem{MultSetType}}) where {MultSetType} = N
   @test !(13^5 in U)
   @test !(13*4289729837 in U)
   @test 5783790198374098 in U
-  @test ambient_ring(U) == R
+  @test ring(U) == R
   W, _ = localization(U)
-  @test base_ring(W) == ambient_ring(U)
+  @test base_ring(W) == ring(U)
   @test inverted_set(W) == U
   a = W(4, 17)
   b = W(4*17)