Generalized skew derivations characterized by acting on zero products

Abstract

Let A be a prime ring such that its symmetric Martindale quotient ring contains a nontrivial idempotent. Then generalized skew derivations of A are characterized by acting on zero products. Precisely, if g;-:A ! A are additive maps such that ae(x)g(y) + -(x)y = 0 for all x;y 2 A with xy = 0 where ae is an automorphism of A, then both g and - are characterized as specific generalized ae-derivations on a nonzero ideal of A. 1. Results Let B be a ring with a subring A. An additive map -:A ! B is called a derivation if -(xy) = -(x)y + x-(y) for all x;y 2 A. In a recent paper Jing, Lu and Li proved the following result (6, Theorem 6): Let B be a standard operator algebra in a Banach space X containing the identity operator I, and -:B ! B be a linear map such that -(AB) = -(A)B + A-(B) for any pair A;B 2 B with AB = 0. Then -(AB) = -(A)B + A-(B) i A-(I)B for all A;B 2 B. Moreover, if in addition -(I) = 0, then - is a derivation. The result says that an additive map on a standard operator algebra is almost a derivation if it satisfies the expansion formula of derivations on pair elements with zero product. Since standard operator algebras involve many idempotents, from this point of view Chebotar, Ke and P.-H. Lee studied maps acting on zero products in the context of prime rings (2). To give its precise statement we first fix some notation. Throughout, unless specially stated, A always denotes a prime ring with center Z, extended centroid C and symmetric Martindale quotient ring Q. Moreover, let