Construction of Subalgebras, Ideals and Quotient Rings

sub<R | L> : AlgMat, List -> AlgMat, Hom(Alg)
Given the matrix algebra R, defined as a subring of Mn(S), construct the subring T of R generated by the elements specified by the list L, where L is a list of one or more items of the following types:
(a)
A sequence of n2 elements of S defining an element of R;
(b)
An element of R;
(c)
A set or sequence of elements of R;
(d)
A subring of R;
(e)
A set or sequence of subrings of R.

Each element or subalgebra specified by the list must belong to the same complete matrix algebra. The subalgebra T will be constructed as a subalgebra of some matrix algebra which contains each of the elements and subalgebras specified in the list.

The generators of T consist of the elements specified by the terms of the list L together with the stored generators for subalgebras specified by terms of the list. Repetitions of an element and occurrences of the identity element are removed (unless T is trivial).

The constructor returns the subalgebra T and the inclusion homomorphism f : T -> R.

ideal<R | L> : AlgMat, List -> AlgMat
Given the matrix algebra R, construct the two-sided ideal I of R generated by the elements of R specified by the list L, where the possibilities for L are the same as for the sub-constructor.
lideal<R | L> : AlgMat, List -> AlgMat
Given the matrix algebra R, construct the left ideal I of R generated by the elements of R specified by the list L, where the possibilities for L are the same as for the sub-constructor.
rideal<R | L> : AlgMat, List -> AlgMat
Given the matrix algebra R, construct the right ideal I of R generated by the elements of R specified in the list L, where the possibilities for L are the same as for the sub-constructor.

Example AlgMat_SubAlgebra (H90E4)

We construct the subalgebra of the matrix algebra A (defined above) that is generated by the first generator. 
> Q := RationalField();
> A := MatrixAlgebra< Q, 3 | [ 1/3,0,0, 3/2,3,0, -1/2,4,3],
>        [ 3,0,0, 1/2,-5,0, 8,-1/2,4] >;
> B := sub< A | A.1 >;
> Dimension(B);
3
> B: Maximal;
Matrix Algebra of degree 3 and dimension 3 with 1 generator
over Rational Field
Generators:
[ 1/3    0    0]
[ 3/2    3    0]
[-1/2    4    3]
Basis:
[1 0 0]
[0 1 0]
[0 0 1]
[    0     0     0]
[    1  16/9     0]
[    0 88/27  16/9]
[   0    0    0]
[   0    0    0]
[   1 16/9    0]
V2.28, 13 July 2023