## Prime Ideal is Irreducible in a Commutative Ring

## Problem 173

Let $R$ be a commutative ring. An ideal $I$ of $R$ is said to be **irreducible** if it cannot be written as an intersection of two ideals of $R$ which are strictly larger than $I$.

Prove that if $\frakp$ is a prime ideal of the commutative ring $R$, then $\frakp$ is irreducible.

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