Bell, H. E.; Daif, M. N. On derivations and commutativity in prime rings. (English) Zbl 0822.16033 Acta Math. Hung. 66, No. 4, 337-343 (1995). Let \(R\) be a prime ring, \(U\) be a right ideal of \(R\), and \(d\) be a nonzero derivation of \(R\). It is shown that each of the following three conditions (i) \([d(x),d(y)] = d([y,x])\) for all \(x,y\in R\), (ii) \([d(x),d(y)] = d([x,y])\) for all \(x,y\in R\), (iii) \(\text{char\,}R\neq 2\) and \(d([x,y]) = 0\) for all \(x,y\in R\), implies that either \(R\) is commutative or \(d^ 2(U) = d(U)^ 2 = \{0\}\). Reviewer: M.Brešar (Maribor) Cited in 4 ReviewsCited in 61 Documents MSC: 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16U80 Generalizations of commutativity (associative rings and algebras) Keywords:commutativity theorems; prime rings; right ideals; derivations PDFBibTeX XMLCite \textit{H. E. Bell} and \textit{M. N. Daif}, Acta Math. Hung. 66, No. 4, 337--343 (1995; Zbl 0822.16033) Full Text: DOI References: [1] H. E. Bell and W. S. Martindale III, Centralizing mappings of semiprime rings,Canad. Math. Bull.,30 (1987), 92–101. · Zbl 0614.16026 · doi:10.4153/CMB-1987-014-x [2] H. E. Bell and L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions,Acta Math. Hungar.,53 (1989), 339–346. · Zbl 0705.16021 · doi:10.1007/BF01953371 [3] H. E. Bell and M. N. Daif, On commutativity and strong commutativity-preserving maps, to appear inCanad. Math. Bull. · Zbl 0820.16031 [4] M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings,Internat. J. Math. and Math. Sci.,15 (1992), 205–206. · Zbl 0746.16029 · doi:10.1155/S0161171292000255 [5] I. N. Herstein, A note on derivations,Canad. Math. Bull. 21 (1978), 369–370. · Zbl 0412.16018 · doi:10.4153/CMB-1978-065-x [6] E. C. Posner, Derivations in prime rings,Proc. Amer. Math. Soc.,8 (1957), 1093–1100. · Zbl 0082.03003 · doi:10.1090/S0002-9939-1957-0095863-0 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.