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Bernstein algebras. (English) Zbl 0597.17014
It is shown that the decomposition of a Bernstein algebra with respect to an idempotent is nothing else but the Peirce decomposition which is known for power associative algebras with idempotent, especially for Jordan algebras with idempotent. Futhermore all idempotents of a Bernstein algebra are principal and primitive. For every Bernstein algebra a necessary and sufficient condition to be a Jordan algebra is given. All trivial Bernstein algebras are Jordan algebras, and beyond this, they are special Jordan algebras. Furthermore all normal Bernstein algebras are Jordan algebras.

MSC:
17D92 Genetic algebras
17A60 Structure theory for nonassociative algebras
17C50 Jordan structures associated with other structures
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