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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.…

## A finite unital ring with no zero divisors is a field

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.21 Solution: [We will assume Wedderburn’s Theorem.] Let $R$ be a finite ring (not necessarily commutative) with…

## The ideal generated by the variable is maximal iff the coefficient ring is a field

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.18 Solution: We first prove a lemma. Lemma: The map $\varphi : R[[x]] \rightarrow R$ given by…

## Prove that a given quotient ring is not an integral domain

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.17 Solution: (1) Evidently, $$\overline{p(x)} = \overline{-x^2-11x+3},$$ $$\overline{q(x)} = \overline{8x^2 - 13x + 5},$$ \overline{p(x)+q(x)} = \overline{7x^2…

## If a nontrivial prime ideal contains no zero divisors, then the ring is an integral domain

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.10 Solution: Let $a,b \in R$ such that $ab = 0$. Since $P$ is a prime ideal…

## In an integral domain, two principal ideals are equal precisely when their generators are associates

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.8 Solution: ($\Rightarrow$) Suppose $(a) = (b)$. Then $a \in (b)$, and we have $a = ub$…