Characterize the units and nilpotent elements of a polynomial ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.33 Solution: (1) Note first that if $a_0$ is a unit in $R$ and $a_i$ nilpotent in…
Ring homomorphisms preserve nilpotency
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.32 Solution: Suppose $x^n = 0$. Then $$\varphi(x)^n = \varphi(x^n) = \varphi(0) = 0,$$ so that $\varphi(x)$…
In a noncommutative ring, the set of nilpotent elements need not be an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.31 Solution: We begin with a lemma. Lemma: Let $R$ be a ring with $1 \neq 0$.…
The nilradical of the quotient of a ring by its nilradical is trivial
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.30 Solution: Suppose $x + \mathfrak{N}(R) \in \mathfrak{N}(R/\mathfrak{N}(R))$. Then for some positive natural number $n$, we have…
The set of nilpotent elements in a commutative ring is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.29 Solution: Let $x,y \in \mathfrak{N}(R)$. Then for some nonnegative natural numbers $n$ and $m$, we have…
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…