By an Halgebra we mean a
nonassociative algebra (not necessarily finitedimensional) over a field in which every
subalgebra is an ideal of the algebra.
In this paper we prove
Main Theorem. Let A be a powerassociative algebra over a field F of
characteristic not 2. A is an Halgebra if and only if A is one of the following;
(1) a onedimensional idempotent algebra;
(2) a zero algebra;
(3) an algebra with basis u_{0},u_{i},i ∈ I (an index set of arbitrary cardinality)
satisfying u_{i}u_{j} = α_{ij}u_{0},α_{ij} ∈ F,i,j ∈ I, all other products zero. Moreover if
J is a finite subset of I, then ∑
_{tj∈J}α_{ij}x_{i}x_{j} is nondegenerate in that
∑
_{i,j∈J}α_{ij}α_{t}α_{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.
