Finite-Dimensional Associative Algebras

These algebras are presented in terms of a basis for a free module Mtogether with a set of structure constants defining the multiplication of these basis elements. It is assumed that we have an echelonization algorithm for M so that standard bases may be constructed for submodules. We shall refer to these algebras as ASC-algebras.

• Creation of algebras in terms of structure constants
• Direct sum
• Arithmetic including Lie bracket operation
• Properties of elements: idempotent, unit, zero-divisor, nilpotent
• Trace and minimal polynomial
• Creation of subalgebras, ideals and quotient algebras
• Ideal arithmetic: Sum, product, powers, intersection
• Centralizer, idealizer
• Characteristic ideals: Centre, commutator ideal, Jacobson radical
• Ideal structure: Maximal (minimal) left, right, two-sided ideals
• Decomposition: Simplicity, semi-simplicity, composition series
• Construction of the (left, right) regular matrix representation
• Lie algebra defined by the Lie product

Functions relating to the ideal structure (Jacobson radical, composition series, maximal and minimal ideals etc) are implemented by applying the module theory machinery to the regular representation of the algebra.