Power-associative algebras in which every subalgebra is an ideal

David Lewis Outcalt

Vol. 20 (1967), No. 3, 481–485

DOI: 10.2140/pjm.1967.20.481

Abstract

By an H-algebra we mean a nonassociative algebra (not necessarily finite-dimensional) over a field in which every subalgebra is an ideal of the algebra.

In this paper we prove

Main Theorem. Let A be a power-associative algebra over a field F of characteristic not 2. A is an H-algebra if and only if A is one of the following;

(1) a one-dimensional idempotent algebra;

(2) a zero algebra;

(3) an algebra with basis u₀,u_i,i ∈ I (an index set of arbitrary cardinality) satisfying u_iu_j = α_iju₀,α_ij ∈ F,i,j ∈ I, all other products zero. Moreover if J is a finite subset of I, then ∑ _tj∈Jα_ijx_ix_j is nondegenerate in that ∑ _i,j∈Jα_ijα_tα_j = 0,α_i,α_j ∈ F,i ∈ J implies α_i = 0,i ∈ J;

(4) direct sums of algebras of types (1), (2), (3) with at most one from each.

Mathematical Subject Classification

Primary: 17.20

Milestones

Received: 5 April 1966

Published: 1 March 1967