Add coefficient_ring for modules and series and improve the docs for base_ring (#2185)

Author: SoongNoonien

docs/src/extending_abstractalgebra.md

@@ -29,6 +29,14 @@ base_ring
 base_ring_type
 ```
 
+If there is a well-defined notion of a coefficient ring (e.g. in the case of
+polynomial rings or modules), then one should implement
+
+```@docs
+coefficient_ring
+coefficient_ring_type
+```
+
 ## Special elements
 
 For rings, one has to extend the following methods:

src/RelSeries.jl

@@ -29,6 +29,10 @@ base_ring_type(::Type{<:SeriesRing{T}}) where T <: RingElement = parent_type(T)
 
 base_ring(R::SeriesRing{T}) where T <: RingElement = R.base_ring::parent_type(T)
 
+coefficient_ring_type(T::Type{<:SeriesRing}) = base_ring_type(T)
+
+coefficient_ring(R::SeriesRing) = base_ring(R)
+
 function is_domain_type(::Type{T}) where {S <: RingElement, T <: SeriesElem{S}}
    return is_domain_type(S)
 end

src/fundamental_interface.jl

@@ -77,7 +77,7 @@ parent_type(T::DataType) = throw(MethodError(parent_type, (T,)))
 @doc raw"""
     base_ring(a)
 
-Return base ring $R$ of given element or parent $a$.
+Return the internal base ring of the given element or parent $a$.
 
 # Examples
 ```jldoctest
@@ -101,7 +101,7 @@ base_ring(x::NCRingElement) = base_ring(parent(x))
 @doc raw"""
     base_ring_type(a)
 
-Return the type of the base ring of the given element, element type, parent or parent type $a$.
+Return the type of the internal base ring of the given element, element type, parent or parent type $a$.
 
 # Examples
 ```jldoctest
@@ -133,9 +133,52 @@ base_ring_type(x::Type{<:ModuleElem}) = base_ring_type(parent_type(x))
 base_ring_type(x::Type{<:Ideal}) = base_ring_type(parent_type(x))
 base_ring_type(T::DataType) = throw(MethodError(base_ring_type, (T,)))
 
-# generic coefficient_ring method
+@doc raw"""
+    coefficient_ring(a)
+
+Return the coefficient ring of the given element or parent $a$.
+
+# Examples
+```jldoctest
+julia> R, x = polynomial_ring(QQ, :x)
+(Univariate polynomial ring in x over rationals, x)
+
+julia> coefficient_ring(x^2+1) == QQ
+true
+
+julia> S, (z,w) = universal_polynomial_ring(QQ, [:z,:w])
+(Universal Polynomial Ring over Rationals, AbstractAlgebra.Generic.UnivPoly{Rational{BigInt}}[z, w])
+
+julia> coefficient_ring(S) == QQ
+true
+```
+"""
+function coefficient_ring end
 coefficient_ring(x::NCRingElement) = coefficient_ring(parent(x))
 
+@doc raw"""
+    coefficient_ring_type(a)
+
+Return the type of the coefficient ring of the given element, element type, parent or parent type $a$.
+
+# Examples
+```jldoctest
+julia> R, x = polynomial_ring(ZZ, :x)
+(Univariate polynomial ring in x over integers, x)
+
+julia> coefficient_ring_type(R) == typeof(coefficient_ring(R))
+true
+
+julia> coefficient_ring_type(zero(R)) == typeof(coefficient_ring(zero(R)))
+true
+
+julia> coefficient_ring_type(typeof(R)) == typeof(coefficient_ring(R))
+true
+
+julia> coefficient_ring_type(typeof(zero(R))) == typeof(coefficient_ring(zero(R)))
+true
+```
+"""
 coefficient_ring_type(x) = coefficient_ring_type(typeof(x))
 coefficient_ring_type(x::Type{<:NCRingElement}) = coefficient_ring_type(parent_type(x))
 coefficient_ring_type(x::Type{<:ModuleElem}) = coefficient_ring_type(parent_type(x))

src/generic/FreeModule.jl

@@ -16,6 +16,10 @@ base_ring_type(::Type{FreeModule{T}}) where T <: NCRingElement = parent_type(T)
 
 base_ring(M::FreeModule{T}) where T <: NCRingElement = M.base_ring::parent_type(T)
 
+coefficient_ring_type(T::Type{<:FreeModule}) = base_ring_type(T)
+
+coefficient_ring(M::FreeModule) = base_ring(M)
+
 elem_type(::Type{FreeModule{T}}) where T <: NCRingElement = FreeModuleElem{T}
 
 parent(m::FreeModuleElem{T}) where T <: NCRingElement = m.parent

src/generic/LaurentSeries.jl

@@ -41,6 +41,14 @@ base_ring(R::LaurentSeriesRing{T}) where T <: RingElement = R.base_ring::parent_
 
 base_ring(R::LaurentSeriesField{T}) where T <: FieldElement = R.base_ring::parent_type(T)
 
+coefficient_ring_type(T::Type{<:LaurentSeriesRing}) = base_ring_type(T)
+
+coefficient_ring_type(T::Type{<:LaurentSeriesField}) = base_ring_type(T)
+
+coefficient_ring(R::LaurentSeriesRing) = base_ring(R)
+
+coefficient_ring(R::LaurentSeriesField) = base_ring(R)
+
 function is_domain_type(::Type{T}) where {S <: RingElement, T <: LaurentSeriesElem{S}}
    return is_domain_type(S)
 end

src/generic/PuiseuxSeries.jl

@@ -57,6 +57,14 @@ base_ring(R::PuiseuxSeriesRing{T}) where T <: RingElement = base_ring(laurent_ri
 
 base_ring(R::PuiseuxSeriesField{T}) where T <: FieldElement = base_ring(laurent_ring(R))
 
+coefficient_ring_type(T::Type{<:PuiseuxSeriesRing}) = base_ring_type(T)
+
+coefficient_ring_type(T::Type{<:PuiseuxSeriesField}) = base_ring_type(T)
+
+coefficient_ring(R::PuiseuxSeriesRing) = base_ring(R)
+
+coefficient_ring(R::PuiseuxSeriesField) = base_ring(R)
+
 @doc raw"""
     max_precision(R::PuiseuxSeriesRing{T}) where T <: RingElement
 

src/generic/QuotientModule.jl

@@ -20,6 +20,10 @@ base_ring_type(::Type{QuotientModule{T}}) where T <: RingElement = parent_type(T
 
 base_ring(N::QuotientModule{T}) where T <: RingElement = N.base_ring::parent_type(T)
 
+coefficient_ring_type(T::Type{<:QuotientModule}) = base_ring_type(T)
+
+coefficient_ring(N::QuotientModule) = base_ring(N)
+
 number_of_generators(N::QuotientModule{T}) where T <: RingElement = length(N.gen_cols)
 
 gens(N::QuotientModule{T}) where T <: RingElement = elem_type(N)[gen(N, i) for i = 1:ngens(N)]

test/generic/FreeModule-test.jl

@@ -11,6 +11,9 @@
    @test elem_type(Generic.FreeModule{elem_type(R)}) == Generic.FreeModuleElem{elem_type(R)}
    @test parent_type(Generic.FreeModuleElem{elem_type(R)}) == Generic.FreeModule{elem_type(R)}
 
+   @test coefficient_ring(M) == R
+   @test coefficient_ring_type(M) == typeof(coefficient_ring(M))
+
    @test isa(M, Generic.FreeModule)
 
    @test isa(M([x, x, x, x, x]), Generic.FreeModuleElem)

test/generic/LaurentSeries-test.jl

@@ -40,10 +40,16 @@ end
    @test elem_type(Generic.LaurentSeriesRing{elem_type(S)}) == Generic.LaurentSeriesRingElem{elem_type(S)}
    @test parent_type(Generic.LaurentSeriesRingElem{elem_type(S)}) == Generic.LaurentSeriesRing{elem_type(S)}
 
+   @test coefficient_ring(T) == S
+   @test coefficient_ring_type(T) == typeof(coefficient_ring(T))
+
    @test elem_type(U) == Generic.LaurentSeriesFieldElem{Rational{BigInt}}
    @test elem_type(Generic.LaurentSeriesField{Rational{BigInt}}) == Generic.LaurentSeriesFieldElem{Rational{BigInt}}
    @test parent_type(Generic.LaurentSeriesFieldElem{Rational{BigInt}}) == Generic.LaurentSeriesField{Rational{BigInt}}
 
+   @test coefficient_ring(U) === QQ
+   @test coefficient_ring_type(U) == typeof(coefficient_ring(U))
+
    @test isa(R, Generic.LaurentSeriesRing)
    @test isa(T, Generic.LaurentSeriesRing)
    @test isa(U, Generic.LaurentSeriesField)

test/generic/PuiseuxSeries-test.jl

@@ -42,10 +42,16 @@ end
    @test elem_type(Generic.PuiseuxSeriesRing{elem_type(S)}) == Generic.PuiseuxSeriesRingElem{elem_type(S)}
    @test parent_type(Generic.PuiseuxSeriesRingElem{elem_type(S)}) == Generic.PuiseuxSeriesRing{elem_type(S)}
 
+   @test coefficient_ring(T) == S
+   @test coefficient_ring_type(T) == typeof(coefficient_ring(T))
+
    @test elem_type(U) == Generic.PuiseuxSeriesFieldElem{Rational{BigInt}}
    @test elem_type(Generic.PuiseuxSeriesField{Rational{BigInt}}) == Generic.PuiseuxSeriesFieldElem{Rational{BigInt}}
    @test parent_type(Generic.PuiseuxSeriesFieldElem{Rational{BigInt}}) == Generic.PuiseuxSeriesField{Rational{BigInt}}
 
+   @test coefficient_ring(U) === QQ
+   @test coefficient_ring_type(U) == typeof(coefficient_ring(U))
+
    @test isa(R, Generic.PuiseuxSeriesRing)
    @test isa(T, Generic.PuiseuxSeriesRing)
    @test isa(U, Generic.PuiseuxSeriesField)