Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.21 Let $X$ be a nonempty set and let $R = \mathcal{P}(X)$ denote the power set of…
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 7.1.18 Let $R$ be a ring. Prove that $\Delta(R) = \{(r,r) \ |\ r \in R \}$…
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 7.1.16 Prove that the only boolean ring that is an integral domain is $\mathbb{Z}/(2)$. Solution: Let $B$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.15 A ring $R$ is called Boolean if $a^2 = a$ for all $a \in R$. Prove…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.14 Let $R$ be a commutative ring and let $x \in R$ be nilpotent – that is,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.13 An element $x \in R$, $R$ a ring, is called nilpotent if $x^m = 0$ for some…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.12 Prove that any subring of a field which contains the identity is an integral domain. Solution:…