The union of a chain of ideals is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.19 Solution: In Exercise 2.1.15, we saw that $S \subseteq R$ is an additive subgroup. To show…
Ring homomorphisms map an identity element to an identity or a zero divisor
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.17 Solution: (1) Suppose $\varphi(1_R) = r$, with $r \neq 1$. First, if $r = 0$, then…
Homomorphic images of ring centers are central
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.16 Solution: Suppose $r \in \varphi[Z(R)]$. Then $r = \varphi(z)$ for some $z \in Z(R)$. Now let…
The Boolean ring of subsets is isomorphic to the ring of characteristic functions
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.15 Solution: Let $\Phi : \mathcal{P}(X) \rightarrow {}^X\mathbb{Z}/(2)$ be the mapping given by $A \mapsto \chi_A$, and…
Embed the Hamiltonian quaternions in a ring of real matrices
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.14 Solution: Define $\varphi : \mathbb{H} \rightarrow M_4(\mathbb{R})$ as follows. $$a+bi+cj+dk \mapsto \begin{bmatrix} a & b &…
Embed the complex numbers in a ring of real matrices
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.13 Solution: Define $$\varphi : \mathbb{C} \rightarrow M_2(\mathbb{R}) by a+bi \mapsto \begin{bmatrix} a & b \\ -b…
Embed a ring of quadratic integers in a ring of matrices
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.12 Solution: We begin with a lemma. Lemma: If $D \in \mathbb{Z}$ is not a perfect square,…
Exhibit a mapping which is a group homomorphism but not a ring homomorphism
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)…
Decide whether or not a subset of Z[x] is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.10 Solution: (1) We claim that this subset $S$ is an ideal. To that end, suppose $\alpha…
Determine whether a subring is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.9 Solution: We have already seen which of these are subrings. (1) Let $S = \{ f…