The characteristic of an integral domain is prime or zero
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.28 Solution: Suppose the characteristic $n$ of $R$ is composite, and that $n = ab$ where $a$…
Every nonzero Boolean ring has characteristic 2
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.27 Solution: Let $R$ be a Boolean ring. Note that $$1+1 = (1+1)^2 = 1+1+1+1,$$ so that…
Basic properties of the characteristic of a ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.26 Solution: (1) We begin by showing that $\varphi(a+b) = \varphi(a) + \varphi(b)$ for nonnegative $b$ by…
The Binomial Theorem holds in any commutative ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.25 Solution: We begin with some lemmas. Recall that ${n \choose k} = \frac{n!}{k!(n-k)!}$, where $n$ is…
The ring homomorphic image of an ideal is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.24 Solution: (1) Let $x,y \in \varphi^\ast[J]$. Now $0 \in J$ and $\varphi(0) = 0$, so that…
A fact about ideals and subrings which intersect trivially
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.22 Solution: By the Second Isomorphism Theorem for rings, we have $$(S+I)/I \cong S/(S \cap I) =…
In a ring, sets of left and right annihilators are one-sided ideals
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.22 Solution: We begin with a definition and some lemmas. Definition: Let $R$ be a ring, $A…
Characterize the two-sided ideals of a matrix ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.21 Solution: First, let $I \subseteq M_n(R)$ be a two-sided ideal. Let $J \subseteq R$ consist of…
The intersection of an ideal and a subring is a subideal, but not necessarily vice versa
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.20 Solution: $I \cap S$ is a subring by Exercise 7.1.4, so it suffices to show absorption.…
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 $R$.…