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## Infinite Boolean rings exist

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.22 Give an example of an infinite boolean ring. Solution: If $X$ is an infinite set, then…

## The powerset of a set is a Boolean ring under intersection and symmetric difference

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.21 Let $X$ be a nonempty set and let $R = \mathcal{P}(X)$ denote the power set of…

## The only Boolean integral domain is Z/(2)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.16 Prove that the only boolean ring that is an integral domain is $\mathbb{Z}/(2)$. Solution: Let $B$…

## Every Boolean ring is commutative

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.15 A ring $R$ is called Boolean if $a^2 = a$ for all $a \in R$. Prove…