On Rings with Polynomial Identity Xn-x=0

Veselin Perić

If $R\not=0$ is an associative ring with the polynomial identity $x^n-x=0$, where $n>1$ is a fixed natural number, then it is well known that $R$ is commutative. It is also known that any anti-inverse ring $R(\not=0)$ satisfies the polynomial identity $x^3-x=0$ [1]. The structure of anti-inverse rings was described in [2]: they are exactly subdirect sums of $GF(2)$'s and $GF(3)$'s. In generalizing the last result, we prove here that a ring $R$ with the polynomial identity $x^n-x=0(>1)$ is a subdirect sum of $GF(p)$'s, where $p^r-1$ divides $n-1$. We also prove again some known results about commutative regular rings.