On the structure of algebraic algebras

Thomas Laffey

Vol. 62 (1976), No. 2, 461–471

DOI: 10.2140/pjm.1976.62.461

Abstract

Let A be an infinite dimensional (associative) algebra over a field F. It is shown that A has an infinite dimensional commutative subalgebra C of one of the following types:

1. C is generated by one element
2. C² = {0}
3. C³ = {0} and C is an ideal of an ideal of A
4. C is generated by mutually orthogonal idempotents
5. C is a field.

A necessary and sufficient condition (in terms of the quadratic forms over F) is obtained for the validity of the statement: Every infinite dimensional nil algebra over F has an infinite dimensional subalgebra B with B² = {0}. An ideal y(A) of an algebra A (analogous to the F.C. subgroup in group theory) is defined and several properties of it are obtained.

Mathematical Subject Classification

Primary: 16A48, 16A48

Milestones

Received: 28 March 1973

Revised: 13 January 1976

Published: 1 February 1976