Every abelian simple group has prime order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.4 Exercise 3.4.1 Solution: Let $G$ be an abelian simple group. Suppose $G$ is infinite. If $x \in G$…
An example of a nilpotent ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.37 Solution: First we prove a lemma. Lemma: Let $R$ be a ring, and let $I_1, I_2,…
Compute the powers of a given ideal in a polynomial ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.36 Solution: We begin with a lemma. Lemma: If $R$ is a commutative ring with 1 and…
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…
Some properties of ideal arithmetic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.34 Solution: (1) We wish to prove the following: (a) $I+J$ is an ideal of $R$ and…
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…