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Submodules, Quotient Modules and Homomorphisms

Submodules, Quotient Modules and Homomorphisms

Subsections

Construction of Submodules

Submodules may be defined for any type of module. However, functions that depend upon membership testing are only implemented for modules over Euclidean Domains (EDs). The conventions defining the presentations of submodules are as follows:

sub<M | L> : ModTupRng, List -> ModTupRng
sub<M | L> : ModMat, List -> ModMat
Given an R-module M, construct the submodule N generated by the elements of M specified by the list L. Each term L_i of the list L must be an expression defining an object of one of the following types:

The generators stored for N consist of the elements specified by terms L_i together with the stored generators for submodules specified by terms of L_i. Repetitions of an element and occurrences of the zero element are removed (unless N is trivial).

The constructor returns the submodule N and the inclusion homomorphism f : N -> M.


Example RMod_Submodule (H42E16)

We construct a submodule of the 4-dimensional vector space over the field of rational function F[x], where F is GF(5).

> P := PolynomialRing(GF(5));                     
> R<x> := FieldOfFractions(P);
> M := RSpace(R, 4);
> N := sub< M | [1, x, 1-x, 0], [1+2*x-x^2, 2*x, 0, 1-x^4 ] >;
> N;
Vector space of degree 4, dimension 2 over Field of Fractions in x over 
Univariate Polynomial Algebra over GF(5)
Generators:
(1    x    4*x + 1    0)
(4*x^2 + 2*x + 1    2*x    0    4*x^4 + 1)
Echelonized basis:
(1  0  3/(x + 4)  (x^3 + x^2 + x + 1) / (x + 4))
(0  1  (4*x^2 + 2*x + 1) / (x^2 + 4*x)  (4*x^3 + 4*x^2 + 4*x + 4) / (x^2 + 4*x))

Construction of Quotient Modules

quo<M | L> : ModTupRng, List -> ModTupRng
quo<M | L> : ModMat, List -> ModMat
Given an R-module M, construct the quotient module P = M/N, where N is the submodule generated by the elements of M specified by the list L. Each term L_i of the list L must be an expression defining an object of one of the following types:

The generators constructed for N consist of the elements specified by terms L_i together with the stored generators for submodules specified by terms of L_i.

The constructor returns the quotient module P and the natural homomorphism f : M -> P.

M / N : ModTupRng, ModTupRng -> ModTupRng
M / N : ModMat, ModMat -> ModMat
Given a submodule N of the R-module M, construct the quotient module of M by N. If r is defined to be Rank(M) - Rank(N), then L is created as a rank r module on the standard basis.
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