On Additive Maps and Commutativity in Rings |
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Authors: | Howard E. Bell Jason Lucier |
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Affiliation: | 1. Department of Mathematics, Brock University St. Catharines, Ontario, Canada, L2S 3A1
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Abstract: | We study additive maps which are skew-commuting or skew-centralizing on appropriate subsets of a ring R; and we investigate commutativity in prime and semiprime rings admitting a nonzero derivation d such that [d(x),d(y)] = 0 for all x,y in some nonzero one-sided ideal. This paper has two main parts. The first, motivated by a recent result of Bre?ar [3] on triviality of skew-commuting additive maps on prime rings, is a study of additive maps which are skew-commuting or skew-centralizing on subsets of certain rings. The second continues a study, begun years ago by Herstein [7], of prime and semiprime rings R admitting a nonzero derivation d such that d(x)d(y) ? d(y)d(x) = 0 for all x, y in a suitably chosen subset of R. |
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