Algebras are viewed as free modules over a ring R with an additional multiplication. There are no a priori conditions imposed on the ring except that it must be unital, but some functions may require that an echelonization algorithm is available for modules over R and sometimes it is also required that R is a field. For example, quotients of algebras can only be constructed over fields, since otherwise the quotient module is not necessarily a free module over R.

The most general way to define an algebra is by structure constants, but for special types of algebras Magma uses more efficient representations.

Subsections

• The Categories of Algebras

At present, Magma contains six main categories of algebras:

• General algebras represented by structure constants: category AlgGen;

• Associative algebras represented by structure constants: category AlgAss;

• Lie algebras represented by structure constants: category AlgLie;

• Group algebras: category AlgGrp with a special type AlgGrpSub for subalgebras of group algebras;

• Matrix algebras: category AlgMat;

• Finitely presented algebras: category AlgFP.