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…
The product of two finitely generated ideals is finitely generated
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.12 Solution: ($\subseteq$) Let $x = \sum_i r_is_i \in IJ$, where $$r_i = \sum_j t_{i,j}a_j$$ and $$s_i…
If a prime ideal contains a product of ideals then it contains one of the factors
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.11 Solution: Suppose $I \not\subseteq P$. Then there exists $a \in I$ such that $a \notin P$.…
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…
A commutative unital ring is a field precisely when the zero ideal is maximal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.4 Solution: ($\Rightarrow$) Suppose $R$ is a field. Let $I \subseteq R$ be an ideal which properly…
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…
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 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…
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