Basic properties of the central product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.12 Solution: (1) Let $\pi : A \times B \rightarrow (A \times B)/Z$ denote the canonical projection,…
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 center of a direct product is the direct product of the centers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.1 Show that the center of a direct product is the direct product of the centers: $$Z(G_1…
A finite group of width two has a trivial center
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.4 Let $G$ be a group. Prove that if $|G| = pq$ for some primes $p$ and…
Characterize the center of a group ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.13 Let $\mathcal{K} = \{k_1, \ldots, k_m \}$ be a conjugacy class in the finite group $G$.…
Exhibit an element in the center of a group ring of finite group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.12 Let $R$ be a ring with $1 \neq 0$, and let $G = \{g_1, \ldots, g_n…
The center of a matrix ring over a commutative ring is precisely the scalar matrices
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.7 Let $R$ be a commutative ring with 1. Prove that the center of the ring $M_n(R)$…
Basic properties of centralizers in a ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.9 Let $R$ be a ring. For a fixed element $a \in R$, define $C_R(a) = \{r…
Compute the center of the Hamiltonian quaternions
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.8 Describe $Z(\mathbb{H}), where \mathbb{H}$ denotes the Hamiltonian Quaternions. Prove that $\{a+bi \ |\ a,b \in \mathbb{R}…
The center of a ring is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.7 The center of a ring $R$ is $$Z(R) = \{ z \in R \ |\ zr…
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