Jordan rings with nonzero socle

Abstract

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