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Transactions of the American Mathematical Society

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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 $ \mathcal{J}$ be a nondegenerate Jordan algebra over a commutative associative ring $ \Phi $ containing $ \tfrac{1}{2}$. Defining the socle $ \mathcal{G}$ of $ \mathcal{J}$ to be the sum of all minimal inner ideals of $ \mathcal{J}$, we prove that $ \mathcal{G}$ is the direct sum of simple ideals of $ \mathcal{J}$. Our main result is that if $ \mathcal{J}$ is prime with nonzero socle, then either (i) $ \mathcal{J}$ is simple unital and satisfies DCC on principal inner ideals, (ii) $ \mathcal{J}$ is isomorphic to a Jordan subalgebra $ \mathcal{J}'$ of the plus algebra $ {A^ + }$ of a primitive associative algebra A with nonzero socle S, and $ \mathcal{J}'$ contains $ {S^ + }$, or (iii) $ \mathcal{J}$ is isomorphic to a Jordan subalgebra $ \mathcal{J}''$ of the Jordan algebra of all symmetric elements H of a. primitive associative algebra A with nonzero socle S, and $ \mathcal{J}''$ contains $ H\, \cap \,S$. Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if $ \mathcal{J}$ is simple then $ \mathcal{J}$ contains a completely primitive idempotent if and only if either $ \mathcal{J}$ is unital and satisfies DCC on principal inner ideals or $ \mathcal{J}$ 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

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