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A note on a theorem of Jacobson related to periodic rings


Authors: D. D. Anderson and P. V. Danchev
Journal: Proc. Amer. Math. Soc. 148 (2020), 5087-5089
MSC (2010): Primary 16D60, 16S34, 16U60
DOI: https://doi.org/10.1090/proc/15246
Published electronically: September 4, 2020
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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.


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Additional 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

DOI: https://doi.org/10.1090/proc/15246
Keywords: Jacobson theorem, Jacobson radical, periodic rings, commutativity.
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
Article copyright: © Copyright 2020 American Mathematical Society