Nonadditive Strong Commutativity Preserving Derivations and Endomorphisms

Abstract

Abstract. Let S be a nonempty subset of a ring R. A map f : R → R is called strong commutativity preserving on S if [f (x), f (y)] = [x, y] for all x, y ∈ S, where the symbol [x, y] denotes xy − yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal ρ of R, then ρ ⊆ Z, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal I ∪ T −1 (I), then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T .In this paper let R always denote an associated ring with the center Z. For a semiprime ring R denote by C and U its extended centroid and maximal right ring of quotients respectively (see [2] for details).Recall that for a nonempty subset S of a ring R, a map f : R → R is called strong commutativity preserving (scp) on S if [f (x), f (y)] = [x, y] for all x, y ∈ S, where the symbol [x, y] denotes xy − yx. The concept of strong commutativity preserving maps was first introduced by Bell and Mason [4]. Bell and Daif [3] proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal ρ of R, then ρ ⊆ Z, the center of R (see also [11,12] due to Liu et al. and [13] by Ma et al. for the results on generalized derivations). In [3] Bell and Daif also obtained that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal I ∪ T −1 (I), then R contains a nonzero central ideal.Brešar and Miers [5] proved that for a semiprime ring R an additive map f : R → R being strong commutativity preserving on R must have the form f (x) = λx + ξ(x), where λ ∈ C, λ 2 = 1, and ξ is an additive map of R into C.