Abstract
Let R be a ring, S a nonempty subset of R and d a derivation on R. A mapping \(f:R\longrightarrow R\) 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) = x 2 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.
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Presented by Claus M. Ringel.
Huang Shuliang is supported by the Natural Science Research Foundation of Anhui Provincial Education Department (No. KJ2010B114).
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Ali, S., Shuliang, H. On Derivations in Semiprime Rings. Algebr Represent Theor 15, 1023–1033 (2012). https://doi.org/10.1007/s10468-011-9271-9
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DOI: https://doi.org/10.1007/s10468-011-9271-9