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…
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…
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…
In an integral domain, two principal ideals are equal precisely when their generators are associates
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.8 Solution: ($\Rightarrow$) Suppose $(a) = (b)$. Then $a \in (b)$, and we have $a = ub$…
In a polynomial ring, the ideal generated by the indeterminate is prime precisely when the coefficient ring is an integral domain
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.7 Solution: We begin with a lemma. Lemma: Let $R$ be a ring. Then $\varphi : R[x]…
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 $a$…
The ring of formal power series over an integral domain is an integral domain
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.4 Prove that if $R$ is an integral domain then the ring $R[[x]]$ of formal power series…
The only Boolean integral domain is Z/(2)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.16 Prove that the only boolean ring that is an integral domain is $\mathbb{Z}/(2)$. Solution: Let $B$…
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