A characterization of potent rings
Glasgow mathematical journal, Tome 65 (2023) no. 2, pp. 324-327
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An associative ring R is called potent provided that for every $x\in R$, there is an integer $n(x)>1$ such that $x^{n(x)}=x$. A celebrated result of N. Jacobson is that every potent ring is commutative. In this note, we show that a ring R is potent if and only if every nonzero subring S of R contains a nonzero idempotent. We use this result to give a generalization of a recent result of Anderson and Danchev for reduced rings, which in turn generalizes Jacobson’s theorem.
Mots-clés :
Boolean ring, idempotent, Jacobson’s theorem, nilradical, potent ring
Oman, Greg. A characterization of potent rings. Glasgow mathematical journal, Tome 65 (2023) no. 2, pp. 324-327. doi: 10.1017/S0017089522000325
@article{10_1017_S0017089522000325,
author = {Oman, Greg},
title = {A characterization of potent rings},
journal = {Glasgow mathematical journal},
pages = {324--327},
year = {2023},
volume = {65},
number = {2},
doi = {10.1017/S0017089522000325},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089522000325/}
}
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