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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A note on a theorem of Jacobson related to periodic rings
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by D. D. Anderson and P. V. Danchev
Proc. Amer. Math. Soc. 148 (2020), 5087-5089
DOI: https://doi.org/10.1090/proc/15246
Published electronically: September 4, 2020

Abstract:

We show that if $R$ is a ring such that for each $x\in R$ there exist two natural numbers $n(x)$ and $m(x)$ of opposite parity with $x^{n(x)}=x^{m(x)}$, then $R$ is commutative. This extends the classical famous theorem of Jacobson [Ann. of Math. 46 (1945), p. 695–707] for commutativity of potent rings.
References
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Bibliographic Information
  • D. D. Anderson
  • Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
  • Email: dan-anderson@uiowa.edu
  • P. V. Danchev
  • Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bonchev” str., bl. 8, 1113 Sofia, Bulgaria
  • MR Author ID: 346948
  • Email: danchev@math.bas.bg; pvdanchev@yahoo.com
  • Received by editor(s): February 24, 2020
  • Published electronically: September 4, 2020
  • Additional Notes: The work of the second named author was supported in part by the Bulgarian National Science Fund under Grant KP-06 N 32/1 of Dec. 07, 2019.
  • Communicated by: Jerzy Weyman
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 5087-5089
  • MSC (2010): Primary 16D60, 16S34, 16U60
  • DOI: https://doi.org/10.1090/proc/15246
  • MathSciNet review: 4163824