On Semiprime Rings with Generalized Derivations
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In this paper, we investigate the commutativity of a semiprime ring R ad-mitting a generalized derivation F with associated derivation D satisfying any one of theproperties: (i) F(x) ◦D(y) = [x, y], (ii) D(x) ◦ F(y) = F[x; y], (iii) D(x) ◦ F(y) = xy, (iv)F(x ◦ y) = [F(x), y] + [D(y), x], and (v) F[x, y] = F(x) ◦ y − D(y) ◦ x for all x, y in someappropriate subsets of R.
Abstract
In this paper, we investigate the commutativity of a semiprime ring R ad-mitting a generalized derivation F with associated derivation D satisfying any one of theproperties: (i) F(x) ◦D(y) = [x, y], (ii) D(x) ◦ F(y) = F[x; y], (iii) D(x) ◦ F(y) = xy, (iv)F(x ◦ y) = [F(x), y] + [D(y), x], and (v) F[x, y] = F(x) ◦ y − D(y) ◦ x for all x, y in someappropriate subsets of R.
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