A Note on Commuting Automorphisms of Rings

Centralizing Mappings of Semiprime Rings

“…Introduction. Let R denote a ring with center Z, and let S be a nonempty subset of R. A mapping F from R to R is called centralizing on S if [x, F(x)] E Z for all x E S; in the special case where [x, F(x)] = 0 for all x E S, the mapping F is described as commuting on S. Over the last fifteen years, several authors [5,7,8,9,10] have proved commutativity theorems for prime rings admitting automorphisms or derivations which are centralizing on appropriate subsets of R. The culminating theorems in this series, due to Mayne [9], assert that if a prime ring R admits either a nonidentity automorphism or a nonzero derivation which is centralizing on some nonzero ideal U of/?, then R is commutative.…”

On Automorphisms and Commutativity in Semiprime Rings

“…[ f (x), x] ∈ Z(R)) for all x ∈ R. The study of commuting and centralizing mappings began in 1955 when Divinsky [11] proved that a simple artinian ring is commutative if it has a commuting non-identity automorphism. In 1970 Luh [27] generalized Divinsky's result to prime rings. In 1976 Mayne [29] showed that a prime ring must be commutative if it possesses a non-identity centralizing automorphism.…”

An Engel Condition With Automorphisms for Left Ideals

“…In 1970, Luh [29] generalized Divinsky's result to prime rings. In 1976, Mayne [31] showed that a prime ring must be commutative if it possesses a non-identity centralizing automorphism.…”