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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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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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