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

## 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…

## Some more properties of ideal arithmetic

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.35 Solution: (1) We show that $I(J+K) = IJ + IK$; the proof of the other equality…

## The set of ideals of a ring is closed under arbitrary intersections

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.18 Solution: (1) In Exercise 7.1.4, we showed that $I \cap J$ is a subring of $R$.…

## The intersection by an abelian normal subgroup is normal in the product

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.20 Let $G$ be a group and $A,B \leq G$ be subgroups such that $A$ is abelian…

## Bounds on the index of an intersection of two subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.10 Let $G$ be a group and let $H,K \leq G$ be subgroups of finite index; say…

## Finite subgroups with relatively prime orders intersect trivially

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.8 Let $G$ be a group and let $H, K \leq G$ be finite subgroups of relatively…

## The intersection of a nonempty collection of subrings is a subring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.4 Prove that the intersection of any nonempty collection of subrings of a ring is also a…

## Basic properties of normalizers with respect to a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.9 Let $G$ be a group, $H \leq G$, and $A \subseteq G$. Define N_H(A) = \{…

## An arbitrary intersection of subgroups is a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.10 Let $G$ be a group. (1) Prove that if $H$ and $K$ are subgroups of $G$,…