A characterization of potent rings

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.