## Every prime ideal in a Boolean ring is maximal

Prove that in a Boolean ring, every prime ideal is maximal.

Let $R$ be a Boolean ring, and let $P \subseteq R$ be a prime ideal. Now $R/P$ is a Boolean ring and an integral domain, so that by a previous theorem, $R/P \cong \mathbb{Z}/(2)$ is a field. Thus $P \subseteq R$ is maximal.