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A remark on the commutativity of certain rings

Author: Ram Awtar
Journal: Proc. Amer. Math. Soc. 41 (1973), 370-372
MSC: Primary 16A70
MathSciNet review: 0327842
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Abstract: In a recent paper [1] Gupta proved that a division ring satisfying the polynomial identity $ x{y^2}x = y{x^2}y$ is commutative. In this note our goal is to prove the following: If $ R$ is a semiprime ring with $ x{y^2}x - y{x^2}y$ central in $ R$, for all $ x,y$ in $ R$, then $ R$ is commutative.

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

  • [1] R. N. Gupta, Nilpotent matrices with invertible transpose, Proc. Amer. Math. Soc. 24 (1970), 572-575. MR 40 #5628. MR 0252408 (40:5628)
  • [2] I. N. Herstein, Topics in ring theory, Univ. of Chicago Press, Chicago, Ill., 1969. MR 42 #6018. MR 0271135 (42:6018)
  • [3] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. MR 20 #2361. MR 0095863 (20:2361)

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Keywords: Prime ring, semiprime ring, inner derivation, subdirect sum
Article copyright: © Copyright 1973 American Mathematical Society

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