It is possible to create a homomorphism between two modules, take the image and kernel of such and verify that these are submodules of the codomain and domain respectively. The Hom--module can also be created as a module of a Dedekind domain.
ModuleBasis: BoolElt Default: true
Return a homomorphism from the module M into the module N as specified by T from which the images of the generators can be inferred. T may be a map between the vector spaces of same degree as M and N, a matrix over the field of fractions or a sequence of vectors. If ModuleBasis is true then the matrix will be taken to be a transformation between the modules and as such will be expected to have size Dimension(M)*Dimension(N) otherwise it will be interpreted as a transformation between the corresponding vector spaces and will be expected to have size Degree(M)*Degree(N).
The module of homomorphisms between the module M and the module N and the map from the hom--module to the collection of maps from M to N, (such that given an element of the hom--module a homomorphism from M to N is returned). The module is over the same Dedekind domain as M and N.
Return true if M is a submodule of N and the map embedding M into N.
The map giving the morphism from the module M to the module N. Either M is a submodule of N, in which case the embedding of M into N is returned, or N is a quotient module of M, in which case the natural epimorphism from M onto N is returned.
> S := [V|[0,1,0], [4,4,0]];
> Mod := Module(S);
> W := KModule(FieldOfFractions(M), 4);
> S := [W|[3, 2, 1, 0]];
> N := Module(S);
> h := hom<Mod -> N | >;
>> h := hom<Mod -> N | >;
^
Runtime error in map< ... >: No images given
> h := hom<Mod -> N | V.1, V.2, V.3>;
>> h := hom<Mod -> N | V.1, V.2, V.3>;
^
Runtime error in map< ... >: An image for each generator is required
> h := hom<Mod -> N | W![3, 2, 1, 0], W![3*(M!F.1 + 1), 2*(M!F.1 + 1),
> M!F.1 + 1, 0] >;
> h(Mod!(4*V.1));
( 4 )
> h(Mod!V![0, 1, 0]);
( x^2 + 1 )
> I := Image(h);
> I;
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x
over Rational Field
Ideal of M
Generator:
1
> K := Kernel(h);
> K;
Integral Module over Maximal Equation Order of F over Univariate Polynomial Ring
in x over Rational Field
Ideal of M
Generator:
1
> IsSubmodule(K, Mod);
true Mapping from: ModDed: K to ModDed: Mod
> H, m := Hom(Mod, N);
> H; m;
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x over
Rational Field
generated by: (in echelon form)
Ideal of M
Generator:
1/4 * ( 1 0 )
Ideal of M
Generator:
1 * ( 0 1 )
Mapping from: ModDed: H to Power Structure of Map given by a rule [no inverse]
> m(H![5, 20]);
Mapping from: ModDed: Mod to ModDed: N
using
[5]
[20]