A note on a theorem of Jacobson related to periodic rings
HTML articles powered by AMS MathViewer
- by D. D. Anderson and P. V. Danchev PDF
- Proc. Amer. Math. Soc. 148 (2020), 5087-5089 Request permission
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
- N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math. (2) 46 (1945), 695–707. MR 14083, DOI 10.2307/1969205
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
- 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