SOME IDENTITIES INVOLVING MULTIPLICATIVE (GENERALIZED) (α,1)-DERIVATIONS IN SEMIPRIME RINGS

Let R be a semiprime ring, I a nonzero ideal of R and ff be an automorphism of R. A map F : R -> R is said to be a multiplicative (generalized) (alpha, 1)-derivation associated with a map d : R -> R such that F(xy) = F(x)alpha(y) + xd(y), for all x, y 2 is an element of R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: (i) F [x;, y] +/- alpha [x, y] = 0; (ii) F (x circle y) +/- alpha (x circle y) = 0; (iii) F [x, y] = [F(x), y](alpha, 1); (iv) F [x; y] = (F(x) circle y)(alpha,1), (v) F (x circle y) = [F(x), y](alpha,1) and (vi) F (x circle y) = (F(x) circle y)(alpha,1), for all x, y is an element of I.