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## Exhibit a mapping which is a group homomorphism but not a ring homomorphism

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.11 Solution: We know from calculus that \begin{align*}\varphi(f+g) =&\ \int_0^1 (f+g)(x) dx\\ =&\ \int_0^1 f(x) + g(x)…

## Determine whether or not a subset is an ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.8 Solution: (1) Note that $(1,1) \in D$. However, $(1,0)(1,1) = (1,0) \notin D$. Since $D$ does…

## The additive group of rational numbers has no subgroups of finite index

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.21 Prove that $\mathbb{Q}$ has no proper subgroups of finite index. Deduce that $\mathbb{Q}/\mathbb{Z}$ has no proper…

## Counterexamples regarding one-sided zero divisors and inverses

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.30 Let $A = \prod_\mathbb{N} \mathbb{Z}$ be the direct product of countably many copies of $\mathbb{Z}$. Recall…

## 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…