DERIVABLE MAPS AND GENERALIZED DERIVATIONS

Let A be a unital algebra, M be an A -bimodule, L(A, M) be the set of all linear maps from A to M, and R-A be a relation on A. Amap d. L(A, M) is called derivable on R-A if delta(AB) = delta(A) B+A delta (B) for all (A, B) is an element of R-A. One purpose of this paper is to propose the study of derivable maps on a new, but natural, relation R-A. Moreover, we give a characterization of generalized derivations on M-n(C), the nxn matrix algebra over the complex numbers; specifically, a linear map delta on M-n(C) is a generalized derivation iff there exists an M is an element of M-n (C) such that delta (AB) = delta (A) B+ A delta (B), for all A, B is an element of M-n (C) satisfying AMB = 0; in this case delta(I) = cM, for some c is an element of C.