If you find any mistakes, please make a comment! Thank you.

## 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 $I$;…

## 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…

## 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…

## The product of two finitely generated ideals is finitely generated

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.12 Solution: ($\subseteq$) Let $x = \sum_i r_is_i \in IJ$, where $$r_i = \sum_j t_{i,j}a_j$$ and s_i…