Left multipliers and Jordan ideals in rings with involution. (English) Zbl 1243.16045

Oukhtite, Lahcen

Afr. Diaspora J. Math. 11, No. 1, 24-28 (2011).

Summary: The purpose of this paper is to study left multipliers satisfying certain identities on Jordan ideals of rings with involution. Some well known results characterizing commutativity of prime rings by left multipliers are also extended to Jordan ideals.
Theorem 2.4. Let \(J\) be a nonzero *-Jordan ideal of a 2-torsion free ring \(R\) and let \(F\) be a left multiplier such that \(F([x,y])=[x,y]\) for all \(x,y\in J\). If \(R\) is *-prime, then \(F\) is trivial or \(J\subseteq Z(R)\).

MSC:

16W10	Rings with involution; Lie, Jordan and other nonassociative structures
16W25	Derivations, actions of Lie algebras
16N60	Prime and semiprime associative rings
16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)

Keywords:

rings with involution; *-prime rings; Jordan ideals; left multipliers; commutativity theorems; generalized derivations