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Commuting and centralizing mappings in prime rings


Author: J. Vukman
Journal: Proc. Amer. Math. Soc. 109 (1990), 47-52
MSC: Primary 16A12; Secondary 16A68, 16A72
DOI: https://doi.org/10.1090/S0002-9939-1990-1007517-3
MathSciNet review: 1007517
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Abstract: Let $ R$ be a ring. A mapping $ F:R \to R$ is said to be commuting on $ R$ if $ [F(x),x] = 0$ holds for all $ x \in R$. The main purpose of this paper is to prove the following result, which generalizes a classical result of E. Posner: Let $ R$ be a prime ring of characteristic not two. Suppose there exists a nonzero derivation $ D:R \to R$, such that the mapping $ x \mapsto [D(x),x]$ is commuting on $ R$. In this case $ R$ is commutative.


References [Enhancements On Off] (What's this?)

  • [1] H. E. Bell and W. S. Martindale, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1) (1987), 92-101. MR 879877 (88h:16044)
  • [2] M. Brešar and J. Vukman, Jordan derivations on prime rings. Bull. Austral. Math. Soc. 37 (1988), 321-322. MR 943433 (89f:16049)
  • [3] -, On left derivations and related mappings Proc. Amer. Math. Soc. (to appear). MR 1028284 (91a:16026)
  • [4] -, On some additive mappings in rings with involution, Aequationes Math. 38 (1989), 178-185. MR 1018911 (90j:16076)
  • [5] I. N. Herstein, Jordan derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110. MR 0095864 (20:2362)
  • [6] J. H. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1) (1976), 113-115. MR 0419499 (54:7520)
  • [7] -, Ideals and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 86 (1982), 211-212; Erratum 89 (1983), 198. MR 667275 (83k:16025)
  • [8] -, Centralizing mappings of prime rings, Canad. Math. Bull. 27 (1) (1984), 122-126. MR 725261 (85h:16039)
  • [9] E. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. MR 0095863 (20:2361)
  • [10] J. Vukman, Symmetric bi-derivations on prime and semi-prime rings, Aequationes Math. 38 (1989) 245-254. MR 1018917 (90k:16038)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1007517-3
Keywords: Prime ring, derivation, Jordan derivation, inner derivation, commuting mapping, centralizing mapping
Article copyright: © Copyright 1990 American Mathematical Society

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