Generalized Strong Commutativity Preserving Centralizers of Semiprime Γ- Rings

In this paper, we introduce the concept of generalized strong commutativity (Cocommutativity) preserving right centralizers on a subset of a Γ-ring. And we generalize some results of a classical ring to a gamma ring.

Strong commutativity preserving maps of strictly triangular matrix Lie algebras

Let [Formula: see text] be the Lie algebra consisting of all strictly upper triangular [Formula: see text] matrices over a field [Formula: see text]. An invertible linear map [Formula: see text] on [Formula: see text] is called to be strong commutativity preserving (simply denoted by SCP) if [Formula: see text] for any [Formula: see text]. We show that for [Formula: see text], an invertible linear map [Formula: see text] preserves strong commutativity if and only if there exist [Formula: see text] and a linear function [Formula: see text] with [Formula: see text] such that [Formula: see text], where [Formula: see text] is an inner SCP, [Formula: see text] are extremal SCPs, [Formula: see text] is a central SCP.

Generalized Derivations in Rings with Involution

The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char [Formula: see text]. If R admits a generalized derivation [Formula: see text] associated with a derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text]. Moreover, a related result is also obtained.