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 7.4 Exercise 7.4.25 Solution: Let $P \subseteq R$ be a prime ideal. Now $R/P$ is an integral domain. Let…
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…
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.…
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…
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…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.19 Solution: We begin with a lemma. Lemma: If $R$ is a finite integral domain, then $R$…
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…