DERIVATIONS ON PRIME AND SEMI-PRIME RINGS

Abstract

In this paper we will show that if there exist deriva- tions D, G on a n!-torsion free semi-prime ring R such that the mapping D 2 + G is n-commuting on R, then D and G are both commuting on R. And we shall give the algebraic conditions on a ring that a Jordan derivation is zero. Throughout this paper R will be represent an associative ring with center Z(R). The commutator xy i yx (resp. the Jordan product xy + yx) will be denoted by (x;y) (resp. hx;yi). We make extensive use of the basic identities (xy;z) = (x;z)y + x(y;z), (x;yz) = (x;y)z + y(x;z). Let rad(A) denote the (Jacobson) radical of an algebra A. Recall that R is prime if aRb = (0) implies that either a = 0 or b = 0, and is semi-prime if aRa = (0) implies a = 0. An additive mapping D from R to R is called a derivation if D(xy) = D(x)y + xD(y) holds for all x;y 2 R. A derivation D is inner if there exists a 2 R such that D(x) = (a;x) holds for all x 2 R. And also, an additive mapping D from R to R is called a Jordan derivation if D(x 2 ) = D(x)x + xD(x) holds for all x 2 R. An additive mapping F from R to R is said to be a commuting (resp. centralizing) if (F(x);x) = 0 (resp. (F(x);x) 2 Z(R)) holds for all x 2 R. More generally, for a positive integer n, we define a mapping F to be n-commuting if (F(x);x n ) = 0 for all x 2 R. The underlying idea of our research is Posner's second theorem (7, Theorem 2) which is the beginning of the study concerning centralizing and commuting mappings, which states that the existence of a nonzero