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

sub — Method

sub(m::FPModule{T}, gens::Vector{<: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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sub — Method

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 = free_module(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, Hom: N -> M)

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

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

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 = 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, Hom: N -> V)

Functionality for submodules

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

Basic manipulation

supermodule — Method

supermodule(M::Submodule{T}) where T <: RingElement

Return the module that this module is a submodule of.

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is_submodule — Method

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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is_compatible — Method

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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dim — Method

dim(N::Submodule{T}) where T <: FieldElement

Return the dimension of the given vector subspace.

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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 = M([ZZ(1), ZZ(4)])
(1, 4)

julia> N1, = sub(M, [m, n])
(Submodule over integers with 2 generators and no relations, Hom: N1 -> M)

julia> N2, = sub(M, [m])
(Submodule over integers with 1 generator and no relations, Hom: N2 -> M)

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 = vector_space(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, Hom: N -> V)

julia> dim(V)
2

julia> dim(N)
1

Intersection

intersect — Method

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 = free_module(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, Hom: submodule over integers with 2 generators and no relations -> M)

julia> N2 = sub(M, [m])
(Submodule over integers with 1 generator and no relations, Hom: submodule over integers with 1 generator and no relations -> M)

julia> I = intersect(N1, N2)
Any[]