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## Use Zorn’s Lemma to construct an ideal which maximally does not contain a given finitely generated ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.35 Solution: Let $\mathcal{C}$ denote the set of all ideals in $R$ which do not contain $I$;…

## Not every ideal is prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.34 Solution: (1) First we show that $I$ is an ideal. Let $f,g \in I$; then for…

## Characterization of maximal ideals in the ring of all continuous real-valued functions

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.33 Solution: (1) Let $M \subseteq R$ be a maximal ideal, and suppose $M \neq M_c$ for…

## 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…

## 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.…

## In a finite commutative ring, all prime ideals are maximal

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 $R$…

## 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…

## The complete homomorphic preimage of a prime ideal is a prime ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.13 Solution: (1) By Exercise 7.3.24, $\varphi^\ast[P]$ is an ideal of $R$. Now suppose $ab \in \varphi^\ast[P]$.…

## In a unital ring, if the quotient corresponding to an ideal is a field, then the ideal is maximal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.5 Solution: Suppose we have a two-sided ideal $I$ with $M \subseteq I \subseteq R$. By the…

## A commutative unital ring is a field precisely when the zero ideal is maximal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.4 Solution: ($\Rightarrow$) Suppose $R$ is a field. Let $I \subseteq R$ be an ideal which properly…