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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…

## 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 $I$;…