Let R be a p-ring (chR=p,a prime and a^p=a for all a in R). If R is field then it is isomorphic to Zp.

Question

I want to prove that the ring R has exactly p elements. I am able to show that a Boolean ring(2-ring) is isomorphic to Z2 provided it is field by showing it has exactly 2 elements.

Please help me to prove the equation x^p=x has exactly p roots in R.

Answer 1

Let $F$ be the prime field of $R$. Then $F$ has $p$ elements.

A polynomial of degree $p$ has at most $p$ zeros in a field.

Therefore, $F$ is exactly the set of zeros of $x^p-x$, which by hypothesis is $R$. So $R=F$.