Best Abstract Algebra Textbooks in 2023
Introductory level Algebra by Micheal Artin "Algebra" by Michael Artin is a widely used textbook in the field of abstract algebra. It's known for its clear explanations and rigorous approach…
Ten best Linear Algebra Books in 2023
Best linear algebra books I would suggest starting with a basic level textbook and then put more effort on one or two intermediate level textbooks. Advanced level books may not…
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 $I$;…
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…
Definition and basic properties of the Jacobson radical of an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.32 Solution: We begin with a lemma. Lemma: If $a,b \in \mathbb{Z}$ are relatively prime, then $$(a)…
In a commutative ring, prime ideals are radical
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.31 Solution: We begin with a lemma. Lemma: Let $R$ be a commutative ring and let $I…
Definition and basic properties of the radical of an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.30 Solution: First, we certainly have $I \subseteq \mathsf{rad}(I)$ since for all $a \in I$, $a^1 \in…
Prove that the augmentation ideal of a given group ring is nilpotent
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.29 Solution: We begin with some lemmas. Lemma 1: Let $\pi : G \rightarrow H$ be a…