Strong commutativity preserving generalized derivations on multilinear polynomials

Abstract

Let R be a non-commutative prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C  and let F and G be generalized derivations of R such that F(x)G(y)-F(y)G(x) = [x; y], for all x; y ∈ S, where S is a subset of R. Here we will discuss the following cases:

(a) S = [R;R];

b) S = L, where L is a non-central Lie ideal of R;

(c) S = ƒ(R), where ƒ(R) is the set of all evaluations of a non-central multilinear polynomial ƒ(x1; : : : ; xn) on R.

In all cases, if R does not satisfy s4(x1; : : : ; x4), the standard polynomial identity on 4 non-commuting variables, then there exist s; c ∈ U such that F(x) = xs, G(x) = cx, for all x ∈ R, and sc = 1_C (the unit of C). We also study the semiprime case.

Mathematics Subject Classification (2010): 16W25, 16N60.
Key words: Prime rings, differential identities, generalized derivations.