On Derivations in Semiprime Rings

Let R be a ring, S a nonempty subset of R and d a derivation on R. A mapping \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:R\longrightarrow R$\end{document} is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, our purpose is to produce commutativity results for rings and show that if R is a 2-torsion free semiprime ring and I a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) d(x) ∘ d(y) = x ∘ y (ii) d(x) ∘ d(y) = − (x ∘ y) (iii) d(x) ∘ d(y) = 0 (iv) [d(x),d(y)] = − [x,y] (v) d(x) d(y) = xy (vi) d(x)d(y) = − xy (vii) d(x)d(y) = yx (viii) d(x)d(x) = x2 for all x,y ∈ I. Further, if d(I) ≠ 0, then R has a nonzero central ideal. Finally, some examples are given to demonstrate that the restrictions imposed on the hypotheses of the various results are not superfluous.