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
MathSciNet review:
4163824
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Abstract | References | Similar Articles | Additional Information
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.
- N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math. (2) 46 (1945), 695–707. MR 14083, DOI https://doi.org/10.2307/1969205
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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
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
