Left Multipliers Satisfying Certain Algebraic Identities on Lie Ideals of Rings with Involution

Abstract

In this note we investigate left multipliers satisfying certain algebraic identities on Lie ideals of rings with involution and discuss related results. Moreover we provide examples to show that the assumed restrictions cannot be relaxed.

A Note on Pair of Left Centralizers in Prime Ring with Involution

“…Moreover in [3] some related results involving left centralizers have also been discussed. In [10] Oukhtite established similar problems for certain situations involving left centralizers acting on Lie ideals. Recently Ali and Dar [1] proved that if a prime ring with involution of the second kind such that char(R) = 2 admits a left centralizer T : R → R satisfying any one of the following conditions:…”

“…T (xy) = xT (y)) for all x, y ∈ R. An additive mapping T is called a centralizer in case T is a left and a right centralizer of R. Considerable work has been done on left (resp. right) centralizers in prime and semiprime rings during the last few decades (see for example [3,6,10,11,[14][15][16][17]) where further references can be found. The first result studying the commutativity of prime ring involving a special mapping was due to Divinsky [5], who proved that a simple artinian ring is commutative if it has a commuting non-trivial automorphism.…”