In a Boolean ring, all finitely generated ideals are principal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.24 Solution: We begin with a lemma. Lemma: Let $R$ be a ring. Suppose that for all…
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.…
Every nonzero Boolean ring has characteristic 2
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.27 Solution: Let $R$ be a Boolean ring. Note that $$1+1 = (1+1)^2 = 1+1+1+1,$$ so that…
The Boolean ring of subsets is isomorphic to the ring of characteristic functions
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.15 Solution: Let $\Phi : \mathcal{P}(X) \rightarrow {}^X\mathbb{Z}/(2)$ be the mapping given by $A \mapsto \chi_A$, and…
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…