Modules

Magma supports the following categories of modules and lattices:

ModTupFld

vector spaces (K-spaces)

ModMatFld

K-matrix spaces

ModTupRng

R-spaces

ModMatRng

R-matrix spaces

ModGrp

R[G]-modules, where G is a group

ModMPol

modules over the multivariate polynomial rings K[x1, ..., xn]

Lat

lattices (Z-modules over a subring of the real field)

Note that a distinction is made between modules over a field K and over a non-field ring R.

There is also a distinction made in Magma between modules in reduced mode or in embedded mode. A submodule presented in reduced mode has the same number of components as its rank; a submodule presented in embedded mode has the same number of components as the module from which it originated. The Magma convention for specifying one's choice of presentation is to use an intrinsic name ending with `Module' for a reduced presentation, or with `Space' for an embedded presentation. This will set the mode for the module being created, and for submodules and quotient modules calculated from it later.

Thus reduced modules are created by the functions:

• KModule
• KMatrixModule
• RModule
• RMatrixModule

and embedded modules are created by the functions:

• KSpace or VectorSpace
• KMatrixSpace
• RSpace
• RMatrixSpace

G-modules have reduced presentation. The functions constructing them are GModule and PermutationModule.

The function Hom(M, N) returns the matrix space of all homomorphisms from M to N, where M and N are suitable modules. It allows linear transformations to be investigated.

One may create any ring by creating one of these `full' or `free' magmas and then taking the required submodule or quotient module, as appropriate, with the sub or quo constructor.

The functions VectorSpaceWithBasis (and so on) allow the user to specify a non-standard basis.

Generic(M) returns the full module of which M is a submodule.

The identity element is Zero(M) or M!0 .

Example

> Q := RationalField();
> Q3 := VectorSpace(Q, 3);
> Q4 := VectorSpace(Q, 4);
> v := Q4.2; print v, v[2];
(0 1 0 0)
1

> H34 := Hom(Q3, Q4);
> a := H34 ! [ 2, 0, 1, -1/2,  1, 0, 3/2, 4,  4/5, 6/7, 0, -1/3];
> b := H34 ! [ 1/2, -3, 0, 5,  1/3, 2, 4/5, 0,  5, -1, 5, 7];
> c := H34 ! [ -1, 4/9, 1, -4,  5, -5/6, -3/2, 0,  4/3, 7, 0, 7/9];
> d := H34 ! [ -3, 5, 1/3, -1/2,  2/3, 4, -2, 0,  0, 4, -1, 0];
> print a, a[2, 3];
[   2    0    1 -1/2]
[   1    0  3/2    4]
[ 4/5  6/7    0 -1/3]
3/2

> U := sub< H34 | a, b, c, d >; print U;
KMatrixSpace of 3 by 4 GHom matrices and dimension 4 over Rational Field