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## 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…

## The order of conductor f in the ring of quadratic integers is a subring and has index f

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…

## Basic properties of the countable direct sum of the integers

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…

## The intersection of a nonempty collection of subrings is a subring

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…

## In a subring containing the identity, units are units in the ring

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…

## In a unital ring, the negative of a unit is a unit

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 $R$,…

## In a unital ring, (-1)² = 1

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…