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…
Definition and basic properties of the radical of an ideal
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…
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…
The nilradical of the quotient of a ring by its nilradical is trivial
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…
The set of nilpotent elements in a commutative ring is an ideal
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…