Construction of Subalgebras, Ideals and Quotient Algebras

Subalgebras and Ideals

If the coefficient ring \(R\) of an algebra \(A\) is a Euclidean domain then one may construct submodules and ideals of \(A\) in Magma.

sub< A | L >: AlgGen, List -> AlgGen, Map

Create the subalgebra \(S\) of the algebra \(A\) that is generated by the elements defined by \(L\), where \(L\) is a list of one or more items of the following types:

(a)

An element of \(A\);

(b)

A set or sequence of elements of \(A\);

(c)

A subalgebra or ideal of \(A\);

(d)

A set or sequence of subalgebras or ideals of \(A\).

The constructor returns the subalgebra as an algebra of the same type as \(A\). An exception are group algebras, where the subalgebra is either of type AlgAss or of the special type AlgGrpSub. As well as the subalgebra \(S\) itself, the constructor returns the inclusion homomorphism \(f : S \rightarrow A\).

lideal< A | L >: AlgGen, List -> AlgGen, Map

Create the left ideal \(I\) of the algebra \(A\) generated by the elements defined by \(L\), where \(L\) is a list as for the sub constructor above. The constructor returns the left ideal as an algebra of the same type as \(A\) with the same exception for group algebras as for the sub constructor. As well as the left ideal \(I\) itself, the constructor returns the inclusion homomorphism \(f : I \rightarrow A\).

rideal< A | L >: AlgGen, List -> AlgGen, Map

Create the right ideal \(I\) of the algebra \(A\) generated by the elements defined by \(L\), where \(L\) is a list as for the sub constructor above. The constructor returns the right ideal as an algebra of the same type as \(A\) with the same exception for group algebras as for the sub constructor. As well as the right ideal \(I\) itself, the constructor returns the inclusion homomorphism \(f : I \rightarrow A\).

ideal< A | L >: AlgGen, List -> AlgGen, Map

Create the (two-sided) ideal \(I\) of the algebra \(A\) generated by the elements defined by \(L\), where \(L\) is a list as for the sub constructor above. The constructor returns the right ideal as an algebra of the same type as \(A\) with the same exception for group algebras as for the sub constructor. As well as the ideal \(I\) itself, the constructor returns the inclusion homomorphism \(f : I \rightarrow A\).

Quotient Algebras

If the coefficient ring \(R\) of an algebra \(A\) is a field, then quotient algebras of \(A\) may also be constructed.

quo< A | L >: AlgGen, List -> AlgGen, Map

Create the quotient algebra \(Q = A / I\), where \(I\) is the two-sided ideal of \(A\) generated by the elements of \(A\) specified by the list \(L\), which should satisfy the same conditions as for the sub constructor above. The constructor returns the quotient as a structure constant algebra with degree equal to its dimension. If \(A\) is known to be associative, then \(Q\) is of type AlgAss, otherwise \(Q\) is of type AlgGen. As well as the quotient \(Q\) itself, the constructor returns the natural homomorphism \(f : A \rightarrow Q\).

A / S: AlgGen, AlgGen -> AlgGen

The quotient of the algebra \(A\) by the (two-sided) ideal closure of its subalgebra \(S\).