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Every prime ideal in a Boolean ring is maximal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.23

Solution: 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 Exercise 7.1.16, R/P \cong \mathbb{Z}/(2) is a field. Thus P \subseteq R is maximal.


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