Jordan rings with nonzero socle

Authors:
J. Marshall Osborn and M. L. Racine

Journal:
Trans. Amer. Math. Soc. **251** (1979), 375-387

MSC:
Primary 17C10

DOI:
https://doi.org/10.1090/S0002-9947-1979-0531985-4

MathSciNet review:
531985

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Abstract: Let be a nondegenerate Jordan algebra over a commutative associative ring containing . Defining the socle of to be the sum of all minimal inner ideals of , we prove that is the direct sum of simple ideals of . Our main result is that if is prime with nonzero socle, then either (i) is simple unital and satisfies DCC on principal inner ideals, (ii) is isomorphic to a Jordan subalgebra of the plus algebra of a primitive associative algebra *A* with nonzero socle *S*, and contains , or (iii) is isomorphic to a Jordan subalgebra of the Jordan algebra of all symmetric elements *H* of a. primitive associative algebra *A* with nonzero socle *S*, and contains . Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if is simple then contains a completely primitive idempotent if and only if either is unital and satisfies DCC on principal inner ideals or is isomorphic to the Jordan algebra of symmetric elements of a -simple associative algebra *A* with involution containing a minimal one-sided ideal.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0531985-4

Keywords:
Jordan algebra,
quadratic Jordan algebra,
socle,
prime Jordan algebra,
primitive associative ring with nonzero socle,
minimal inner ideal

Article copyright:
© Copyright 1979
American Mathematical Society