Vol. 62, No. 2, 1976

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On the structure of algebraic algebras

Thomas Laffey

Vol. 62 (1976), No. 2, 461–471

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
Received: 28 March 1973
Revised: 13 January 1976
Published: 1 February 1976
Thomas Laffey
National University of Ireland Dublin