Submodules

AbstractAlgebra allows the construction of quotient modules/spaces of AbstractAlgebra modules. These are given as the quotient of a module and a submodule of that module.

As well as implementing the entire Module interface, AbstractAlgebra submodules also provide the following interface.

Constructors

AbstractAlgebra.QuotientModule — Method.

QuotientModule(m::Module{T}, gens::Vector{ModuleElem{T}}) where T <: RingElement

Return the quotient Q of the module m by the submodule N of m, and a map which is a lift of elements of Q to m.

AbstractAlgebra.QuotientSpace — Method.

QuotientSpace(m::Module{T}, gens::Vector{ModuleElem{T}}) where T <: RingElement

Return the quotient Q of the vector space m by the subvector space N of m, and a map which is a lift of elements of Q to m.

Examples

M = FreeModule(ZZ, 2)

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

N, f = Submodule(M, [m])
Q, g = QuotientModule(M, N)

p = Q([ZZ(3)])
v2 = g(p)

V = VectorSpace(QQ, 2)

m = V([QQ(1), QQ(2)])

N, f = Subspace(V, [m])
Q, g = QuotientSpace(V, N)

Functionality for submodules

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

Basic manipulation

AbstractAlgebra.Generic.supermodule — Method.

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

Return the module that this module is a quotient of.

Examples

M = FreeModule(ZZ, 2)
m = M([ZZ(2), ZZ(3)])
N, g = Submodule(M, [m])
Q, h = QuotientModule(M, N)

supermodule(Q) == M