## 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…

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…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.23 Let $D$ be a squarefree integer, and let $O$ be the ring of integers in the…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.20 Let $R$ be the collection of sequences $(a_i)$ of integers, indexed by $\mathbb{N}$, such that all…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.19 Let $I$ be a nonempty index set and let $R_i$ be a ring for each $i…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.17 Let $R$ and $S$ be rings. Prove that the direct product $R \times S$ is a…

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}…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.4 Prove that the intersection of any nonempty collection of subrings of a ring is also a…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.3 Let $R$ be a ring with identity and let $S \subseteq R$ be a subring containing…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.2 Let $R$ be a ring with 1. Prove that if $u$ is a unit in $R$,…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.1 Let $R$ be a ring with 1. Show that $(-1)^2 = 1$ in $R$. Solution: Let…