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quotient_module.md

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CurrentModule = AbstractAlgebra
DocTestSetup = quote
    using AbstractAlgebra
end

Quotient modules

AbstractAlgebra allows the construction of quotient modules/spaces of AbstractAlgebra modules over euclidean domains. These are given as the quotient of a module by a submodule of that module.

We define two quotient modules to be equal if they are quotients of the same module $M$ by two equal submodules.

Generic quotient module type

AbstractAlgebra implements the generic quotient module type Generic.QuotientModule{T} where T is the element type of the base ring, in src/generic/QuotientModule.jl.

Elements of generic quotient modules have type Generic.QuotientModuleElem{T}.

Abstract types

Quotient module types belong to the FPModule{T} abstract type and their elements to FPModuleElem{T}.

Constructors

quo(M::FPModule{T}, v::Generic.Submodule{T}) where T <: RingElement

Note that a preimage of the canonical projection can be obtained using the preimage function described in the section on module homomorphisms. Note that a preimage element of the canonical projection is not unique and has no special properties.

Examples

julia> M = free_module(ZZ, 2)
Free module of rank 2 over integers

julia> m = M([ZZ(1), ZZ(2)])
(1, 2)

julia> N, f = sub(M, [m])
(Submodule over integers with 1 generator and no relations, Hom: submodule over integers with 1 generator and no relations -> free module of rank 2 over integers)

julia> Q, g = quo(M, N)
(Quotient module over integers with 1 generator and no relations, Hom: free module of rank 2 over integers -> quotient module over integers with 1 generator and no relations)

julia> p = M([ZZ(3), ZZ(1)])
(3, 1)

julia> v2 = g(p)
(-5)

julia> V = vector_space(QQ, 2)
Vector space of dimension 2 over rationals

julia> m = V([QQ(1), QQ(2)])
(1//1, 2//1)

julia> N, f = sub(V, [m])
(Subspace over rationals with 1 generator and no relations, Hom: subspace over rationals with 1 generator and no relations -> vector space of dimension 2 over rationals)

julia> Q, g = quo(V, N)
(Quotient space over rationals with 1 generator and no relations, Hom: vector space of dimension 2 over rationals -> quotient space over rationals with 1 generator and no relations)

Functionality for submodules

In addition to the Module interface, AbstractAlgebra submodules implement the following functionality.

Basic manipulation

supermodule(M::Generic.QuotientModule{T}) where T <: RingElement

dim(N::Generic.QuotientModule{T}) where T <: FieldElement

Examples

julia> M = free_module(ZZ, 2)
Free module of rank 2 over integers

julia> m = M([ZZ(2), ZZ(3)])
(2, 3)

julia> N, g = sub(M, [m])
(Submodule over integers with 1 generator and no relations, Hom: submodule over integers with 1 generator and no relations -> free module of rank 2 over integers)

julia> Q, h = quo(M, N)
(Quotient module over integers with 2 generators and relations:
[2 3], Hom: free module of rank 2 over integers -> quotient module over integers with 2 generators and relations:
[2 3])

julia> supermodule(Q) == M
true

julia> V = vector_space(QQ, 2)
Vector space of dimension 2 over rationals

julia> m = V([QQ(1), QQ(2)])
(1//1, 2//1)

julia> N, f = sub(V, [m])
(Subspace over rationals with 1 generator and no relations, Hom: subspace over rationals with 1 generator and no relations -> vector space of dimension 2 over rationals)

julia> Q, g = quo(V, N)
(Quotient space over rationals with 1 generator and no relations, Hom: vector space of dimension 2 over rationals -> quotient space over rationals with 1 generator and no relations)

julia> dim(V)
2

julia> dim(Q)
1