Submodules

AbstractAlgebra allows the construction of submodules/subvector spaces of AbstractAlgebra modules over euclidean domains. These are given as the submodule generated by a finite list of elements in the original module.

We define two submodules to be equal if they are (transitively) submodules of the same module $M$ and their generators generate the same set of elements.

Generic submodule type

AbstractAlgebra implements a generic submodule type Generic.Submodule{T} where T is the element type of the base ring in src/generic/Submodule.jl. See src/generic/GenericTypes.jl for more details of the type definition.

Elements of a generic submodule have type Generic.SubmoduleElem{T}.

Abstract types

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

Constructors

AbstractAlgebra.subMethod
sub(m::AbstractAlgebra.FPModule{T}, gens::Vector{<:AbstractAlgebra.FPModuleElem{T}}) where T <: RingElement

Return the submodule of the module m generated by the given generators, given as elements of m.

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AbstractAlgebra.subMethod
sub(m::Module{T}, subs::Vector{<:Generic.Submodule{T}}) where T <: RingElement

Return the submodule S of the module m generated by the union of the given submodules of $m$, and a map which is the canonical injection from S to m.

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Note that the preimage of the canonical injection can be obtained using the preimage function described in the section on module homomorphisms. As the canonical injection is injective, this is unique.

Examples

julia> M = FreeModule(ZZ, 2)
Free module of rank 2 over Integers

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

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

julia> N, f = sub(M, [m, n])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 2 over Integers)

julia> v = N([ZZ(3), ZZ(4)])
(3, 4)

julia> v2 = f(v)
(3, 26)

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 = V([QQ(2), QQ(-1)])
(2//1, -1//1)

julia> N, f = sub(V, [m, n])
(Subspace over Rationals with 2 generators and no relations
, Module homomorphism with
Domain: Subspace over Rationals with 2 generators and no relations

Codomain: Vector space of dimension 2 over Rationals)

Functionality for submodules

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

Basic manipulation

AbstractAlgebra.Generic.is_submoduleMethod
is_submodule(M::AbstractAlgebra.FPModule{T}, N::AbstractAlgebra.FPModule{T}) where T <: RingElement

Return true if $N$ was constructed as a submodule of $M$. The relation is taken transitively (i.e. subsubmodules are submodules for the purposes of this relation, etc). The module $M$ is also considered a submodule of itself for this relation.

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AbstractAlgebra.Generic.is_compatibleMethod
is_compatible(M::AbstractAlgebra.FPModule{T}, N::AbstractAlgebra.FPModule{T}) where T <: RingElement

Return true, P if the given modules are compatible, i.e. that they are (transitively) submodules of the same module, P. Otherwise return false, M.

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Examples

julia> M = FreeModule(ZZ, 2)
Free module of rank 2 over Integers

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

julia> n = M([ZZ(1), ZZ(4)])
(1, 4)

julia> N1, = sub(M, [m, n])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 2 over Integers)

julia> N2, = sub(M, [m])
(Submodule over Integers with 1 generator and no relations
, Module homomorphism with
Domain: Submodule over Integers with 1 generator and no relations

Codomain: Free module of rank 2 over Integers)

julia> supermodule(N1) == M
true

julia> is_compatible(N1, N2)
(true, Free module of rank 2 over Integers)

julia> is_submodule(N1, M)
false


julia> V = VectorSpace(QQ, 2)
Vector space of dimension 2 over Rationals

julia> m = V([QQ(2), QQ(3)])
(2//1, 3//1)

julia> N, = sub(V, [m])
(Subspace over Rationals with 1 generator and no relations
, Module homomorphism with
Domain: Subspace over Rationals with 1 generator and no relations

Codomain: Vector space of dimension 2 over Rationals)

julia> dim(V)
2

julia> dim(N)
1

Intersection

Base.intersectMethod
Base.intersect(M::FPModule{T}, N::FPModule{T}) where T <: RingElement

Return the intersection of the modules $M$ as a submodule of $M$. Note that $M$ and $N$ must be (constructed as) submodules (transitively) of some common module $P$.

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Examples

julia> M = FreeModule(ZZ, 2)
Free module of rank 2 over Integers

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

julia> n = M([ZZ(1), ZZ(4)])
(1, 4)

julia> N1 = sub(M, [m, n])
(Submodule over Integers with 2 generators and no relations
, Module homomorphism with
Domain: Submodule over Integers with 2 generators and no relations

Codomain: Free module of rank 2 over Integers)

julia> N2 = sub(M, [m])
(Submodule over Integers with 1 generator and no relations
, Module homomorphism with
Domain: Submodule over Integers with 1 generator and no relations

Codomain: Free module of rank 2 over Integers)

julia> I = intersect(N1, N2)
Union{AbstractAlgebra.Generic.ModuleHomomorphism{BigInt}, AbstractAlgebra.Generic.Submodule{BigInt}}[]