A ring in which all prime ideals are maximal

#ring_theory

Let $R$ be a commutative ring with identity. Suppose that for every $a \in R$ there is an integer $n \geq 2$ such that $a^{n} = a$ . Show that every prime ideal of $R$ is maximal.

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Let $R$ be a commutative ring with identity.  Suppose that for every $a\in R$ there is an integer $n\geq 2$ such that $a^n=a$. Show that every prime ideal of $R$ is maximal.