## Jordan rings with nonzero socle

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- by J. Marshall Osborn and M. L. Racine
- Trans. Amer. Math. Soc.
**251**(1979), 375-387 - DOI: https://doi.org/10.1090/S0002-9947-1979-0531985-4
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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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*Peirce ideals in Jordan algebras*(to appear).

## Bibliographic Information

- © Copyright 1979 American Mathematical Society
- 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