A Study of Applications and their Involution of left ∗-Centralizers and ∗-Mappings in Rings

Exploring Properties of Rings and their Mappings

Abstract

The commutativity of a ring and the existence of certain specific types of maps on R. The firstresult in this direction is due to prove that a simple artinian ring is commutative if it has a commuting nontrivialautomorphisms. This result was subsequently refined and extended by a number of authors invarious directions (see for example, where further references can be looked). In the year 1957, Posnerproved that the existence of a nonzero centralizing derivation on a prime ring forces the ring to becommutative. Besides proving some other results on prime and semiprime rings with involution, themain objective of this study is to present a ∗ -version of Posner’s theorem. The purpose of the study is toprove that a prime ring R of characteristic not two with a nonzero derivation d satisfying d(x)d(y) =d(y)d(x) for all x, y ∈ R, must be commutative. Further, Daif showed that if a 2-torsion free semiprime ringR admits a derivation d such that d(x)d(y) = d(y)d(x) for all x, y ∈ I, where I is a nonzero ideal of R and d isnonzero on I, then R contains a nonzero central ideal. Further this result was extended by many authors(viz. where further references can be found).