The nilradical of a commutative ring is contained in every prime ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.26 Solution: Let $P \subseteq R$ be a prime ideal, and let $x \in R$ be nilpotent…
A sufficient condition for the ring property that every prime ideal is maximal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.25 Solution: Let $P \subseteq R$ be a prime ideal. Now $R/P$ is an integral domain. Let…
In a Boolean ring, all finitely generated ideals are principal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.24 Solution: We begin with a lemma. Lemma: Let $R$ be a ring. Suppose that for all…
Every prime ideal in a Boolean ring is maximal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.23 Solution: Let R be a Boolean ring, and let P \subseteq R be a prime ideal.…
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…
A nonzero finite commutative ring with no zero divisors is a field
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.20 Solution: Let $R$ be a finite commutative ring with no zero divisors. Now let $I \subseteq…
In a finite commutative ring, all prime ideals are maximal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.19 Solution: We begin with a lemma. Lemma: If $R$ is a finite integral domain, then $R$…
The ideal generated by the variable is maximal iff the coefficient ring is a field
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.18 Solution: We first prove a lemma. Lemma: The map $\varphi : R[[x]] \rightarrow R$ given by…
Prove that a given quotient ring is not an integral domain
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.17 Solution: (1) Evidently, $$\overline{p(x)} = \overline{-x^2-11x+3},$$ $$\overline{q(x)} = \overline{8x^2 - 13x + 5},$$ $$\overline{p(x)+q(x)} = \overline{7x^2…
Compute in a quotient of a polynomial ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.16 Solution: (1) Evidently, $$\overline{g(x)} = \overline{-2x^3 + 25936x + 3}.$$ (2) Note that $$x^4-16 = (x+2)(x-2)(x^2+4).$$…