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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:
- If M has been created using the function RSpace, then
every submodule of M is given in terms of a generating set
consisting of elements of M, i.e. by means of an embedded generating
set.
- If M has been created using the function RModule, then
every submodule of M is given in terms of a reduced basis.
sub<M | L> : ModMatRng, List -> ModMatRng
Given an R-module M, construct the submodule N generated by the
elements of M specified by the list L. Each term Li of the list
L must be an expression defining an object of one of the following
types:
- (a)
- A sequence of n elements of R defining an element of M;
- (b)
- A set or sequence whose terms are elements of M;
- (c)
- A submodule of M;
- (d)
- A set or sequence whose terms are submodules of M.
The generators stored for N consist of the elements specified by
terms Li together with the stored generators for submodules specified
by terms of Li. 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.
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))
The following operations are only available for submodules of
R(n), HomR(M, N) and R[G], where R is a Euclidean
Domain. If the modules involved are R[G]-modules, the operators refer
to the underlying R-module.
u in M : ModMatRngElt, ModMatRng -> BoolElt
Returns true if the element u lies in the R-module M, where u and M
belong to the same R-module.
u notin M : ModMatRngElt, ModMatRng -> BoolElt
Returns true if the element u does not lie in the R-module M, where u
and M belong to the same R-module.
N subset M : ModMatRng, ModMatRng -> BoolElt
Returns true if the R-module N is contained in the R-module M, where
M and N belong to a common R-module.
N notsubset M : ModMatRng, ModMatRng -> BoolElt
Returns true if the R-module N is not contained in the R-module M, where
M and N belong to a common R-module.
M eq N : ModMatRng, ModMatRng -> BoolElt
Returns true if the R-modules N and M are equal, where N and M
belong to a common R-module.
M ne N : ModMatRng, ModMatRng -> BoolElt
Returns true if the R-modules N and M are not equal, where N and M
belong to a common R-module.
The following operations are only available for submodules of
R(n), HomR(M, N) and R[G], where R is a Euclidean
Domain. If the modules involved are R[G]-modules, the operators refer
to the underlying R-module.
M + N : ModMatRng, ModMatRng -> ModMatRng
Sum of the submodules M and N, where M and N belong to a
a common R-module.
M meet N : ModMatRng, ModMatRng -> ModMatRng
Intersection of the submodules M and N, where M and N belong
to a common R-module.
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