## In a commutative ring, prime ideals are radical

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.31 Solution: We begin with a lemma. Lemma: Let $R$ be a commutative ring and let $I…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.31 Solution: We begin with a lemma. Lemma: Let $R$ be a commutative ring and let $I…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.30 Solution: First, we certainly have $I \subseteq \mathsf{rad}(I)$ since for all $a \in I$, $a^1 \in…

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…

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 $b \in R$,…

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…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.30 Solution: Suppose $x + \mathfrak{N}(R) \in \mathfrak{N}(R/\mathfrak{N}(R))$. Then for some positive natural number $n$, we have…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.29 Solution: Let $x,y \in \mathfrak{N}(R)$. Then for some nonnegative natural numbers $n$ and $m$, we have…