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## The set of prime ideals of a commutative ring contains inclusion-minimal elements

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.36 Solution: Let $\mathcal{P}$ denote the set of prime ideals. Note that $\mathcal{P}$ is partially ordered by…

## Definition and basic properties of the Jacobson radical of an ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.32 Solution: We begin with a lemma. Lemma: If $a,b \in \mathbb{Z}$ are relatively prime, then (a)…

## An ideal which is finitely generated by nilpotent elements is nilpotent

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.28 Solution: Let $x,y \in \mathfrak{N}(R)$. Then for some nonnegative natural numbers $n$ and $m$, we have…

## Constructing units from nilpotent elements in a commutative ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.27 Solution: By Exercise 7.3.29, $\mathfrak{N}(R)$ is an ideal of $R$. Thus for all $b \in R$,…

## The nilradical of a commutative ring is contained in every prime ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.26 Solution: Let $P \subseteq R$ be a prime ideal, and let $x \in R$ be nilpotent…

## A sufficient condition for the ring property that every prime ideal is maximal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.25 Solution: Let $P \subseteq R$ be a prime ideal. Now $R/P$ is an integral domain. Let…

## 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 Wedderburnâ€™s Theorem.] Let $R$ be a finite ring (not necessarily commutative) with…

## A nonzero finite commutative ring with no zero divisors is a field

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.20 Solution: Let $R$ be a finite commutative ring with no zero divisors. Now let \$I \subseteq…