Centralizing Mappings of Semiprime Rings

Abstract

Abstract Let R be a ring with center Z, and S a nonempty subset of R . A mapping F from R to R is called centralizing on S if [ x, F(x) ] ∊ Z for all x ∊ S . We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.