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## A finite unital ring with no zero divisors is a field

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.21 Solution: [We will assume Wedderburn’s Theorem.] Let $R$ be a finite ring (not necessarily commutative) with…

## If a nontrivial prime ideal contains no zero divisors, then the ring is an integral domain

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.10 Solution: Let $a,b \in R$ such that $ab = 0$. Since $P$ is a prime ideal…

## Ring homomorphisms map an identity element to an identity or a zero divisor

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.17 Solution: (1) Suppose $\varphi(1_R) = r$, with $r \neq 1$. First, if $r = 0$, then…

## Characterization of zero divisors in a polynomial ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.2 Let $R$ be a commutative ring with $1 \neq 0$. Let $p(x) = \sum_{i=0}^n a_ix^i$ be…

## Counterexamples regarding one-sided zero divisors and inverses

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.30 Let $A = \prod_\mathbb{N} \mathbb{Z}$ be the direct product of countably many copies of $\mathbb{Z}$. Recall…

## Basic properties of left and right units and left and right zero divisors

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.28 Let $R$ be a ring with $1 \neq 0$. A nonzero element $a \in R$ is…

## Basic properties of nilpotent ring elements

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.14 Let $R$ be a commutative ring and let $x \in R$ be nilpotent – that is,…