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 \(M_n(S)\), 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 \(n^2\) 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 \(f : T \rightarrow R\).
- 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: Sub Algebra (ex-3c6faa)#
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]