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.

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