Vol. 20, No. 3, 1967

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Power-associative algebras in which every subalgebra is an ideal

David Lewis Outcalt

Vol. 20 (1967), No. 3, 481–485

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,ui,i I (an index set of arbitrary cardinality) satisfying uiuj = αiju0ij F,i,j I, all other products zero. Moreover if J is a finite subset of I, then tjJαijxixj is nondegenerate in that i,jJαijαtαj = 0ij F,i J implies αi = 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
Received: 5 April 1966
Published: 1 March 1967
David Lewis Outcalt