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Semi-prime Bernstein algebras. (English) Zbl 0643.17018
Let $$A$$ be a nonassociative algebra over a field $$K$$. $$A$$ is called semiprime if it satisfies the following condition: If $$I$$ is an ideal of $$A$$ and $$I^2=0$$, then $$I=0$$. $$A$$ is called a Bernstein algebra if $$A$$ is commutative and has a nontrivial algebra homomorphism $$w: A\to K$$ which satisfies the identity $$x^2x^2=w(x)^2x^2$$ for all $$x\in A$$. The authors prove that: (i) a semiprime Bernstein algebra is a Jordan algebra, (ii) a finitely generated semiprime Bernstein algebra is a field, and (iii) if $$A$$ is a finitely generated Bernstein algebra, then the kernel of $$w$$ is solvable. The proofs require characteristic different from two.
Reviewer: I.R.Hentzel

##### MSC:
 17D92 Genetic algebras 17A30 Nonassociative algebras satisfying other identities 17C99 Jordan algebras (algebras, triples and pairs)
##### Keywords:
semiprime Bernstein algebra; Jordan algebra
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##### References:
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