# 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

quoMethod
quo(m::FPModule{T}, subm::FPModule{T}) where T <: RingElement

Return the quotient M of the module m by the module subm (which must have been (transitively) constructed as a submodule of m or be m itself) along with the canonical quotient map from m to M.

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 = FreeModule(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 = VectorSpace(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

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

Return the module that this module is a quotient of.

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

Return the dimension of the given vector quotient space.

Examples

julia> M = FreeModule(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 = VectorSpace(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