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…
Compute in a group ring of symmetric group over Z/(3)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.11 Consider the following elements of the group ring $\mathbb{Z}/(3)[S_3]$: $$\alpha = 1(2\ 3) + 2(1\ 2\…
Compute in a group ring of symmetric group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.10 Consider the following elements of the integral group ring $\mathbb{Z}[S_3]$: $$\alpha = 3(1\ 2) - 5(2\…
Compute in a group ring over dihedral group of order 8
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.9 Let $\alpha = r + r^2 - 2s$ and $\beta = -3r^2 + rs$ be elements…
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 $M_n(R)$…
Definition and properties of matrices with a single nonzero entry
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.6 Solution: (1) By definition, $E_{i,j}A = [c_{p,q}]$, where $$c_{p,q} = \sum_{k=1}^n e_{p,k}a_{k,q}.$$ Note that if $p…
The set of all formal Laurent series is a ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.5 Let $F$ be a field and define the ring $F((x))$ of formal Laurent series with coefficients…
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 set of formal power series is a ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.3 Let $R$ be a ring. Define the set $R[[x]]$ of formal power series in the indeterminate…
Characterization of zero divisors in a polynomial ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.2 Let $R$ be a commutative ring with $1 \neq 0$. Let $p(x) = \sum_{i=0}^n a_ix^i$ be…