Let $R$ be a commutative ring of characteristic $p$ (prime). It is evident that $a\mapsto a^p$ is an endomorphism of $R$. But is it an automorphism?
This is the same as asking whether the equation $x^p = 0$ has only $x =0$ as a solution. It doesn't look true to me. If $R$ is a domain then clearly this is true. What about non-domains?
Maybe this is very easy but I'm a beginner and I don't have much examples of rings. All the rings I know either satisfy this or do not have characteristic $p$.