Abstract
In this paper we study the semiprime case. That is, we consider Bernstein algebras that do not have nonzero nilpotent ideals of index two. We prove that any such algebra is Jordan. Furthermore, under the condition that the algebra is finitely generated, we show that it must be a field. The proofs require characteristic different from two. Our work implies that nearly all (finitely generated) Bernstein algebras possess nonzero ideals which are nilpotent of index two. The only ones which do not are the fields.
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