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 | References | Similar Articles | Additional Information
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