NOTES ON LEFT IDEALS OF SEMIPRIME RINGS WITH MULTIPLICATIVE GENERALIZED (alpha,alpha)-DERIVATIONS

Let R be a 2-torsion free semiprime ring, I a nonzero left ideal of R, alpha an automorphism on R and F a multiplicative (generalized) (alpha,alpha)-derivation of R. This note we gave the description of commutativity of semiprime rings with help of some identities involving a multiplicative generalized (alpha,alpha)-derivation and a nonzero left ideal of R.