Let $R$ be a commutative ring with unity. For every $x\in R$ there exists $n_i>1$ such that $x^{n_i}=x$. Then every prime ideal of $R$ is maximal.
Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?