## Not all ideals are prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.9 Solution: First we show that $I$ is an ideal. To That end, let $f,g \in I$.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.9 Solution: First we show that $I$ is an ideal. To That end, let $f,g \in I$.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.7 Solution: (1) $Q_8$ is a subgroup of $S_8$ via the left regular representation. (2) Now suppose…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.5 Solution: Write $Z_4 = \langle x \rangle$ and consider $\langle (i,x) \rangle = \{ (i,x), (-1,x^2),…

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 $1 \neq 0$.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.20 Solution: $I \cap S$ is a subring by Exercise 7.1.4, so it suffices to show absorption.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.11 Solution: We know from calculus that \begin{align*}\varphi(f+g) =&\ \int_0^1 (f+g)(x) dx\\ =&\ \int_0^1 f(x) + g(x)…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.8 Solution: (1) Note that $(1,1) \in D$. However, $(1,0)(1,1) = (1,0) \notin D$. Since $D$ does…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.6 Decide which of the following are ring homomorphisms from $M_2(\mathbb{Z}) to \mathbb{Z}$. (1) $\begin{bmatrix} a &…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.2 Prove that the rings $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic. Proof: In $\mathbb{Q}[x]$, $f(x)+f(x)=g(x)$ has a…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.1 Prove that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic. Solution: Suppose $\varphi : 2\mathbb{Z} \rightarrow…