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## Not every ideal is prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.34 Solution: (1) First we show that $I$ is an ideal. Let $f,g \in I$; then for…

## Characterization of maximal ideals in the ring of all continuous real-valued functions

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.33 Solution: (1) Let $M \subseteq R$ be a maximal ideal, and suppose $M \neq M_c$ for…

## Not all ideals are prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.9 Solution: First we show that $I$ is an ideal. To That end, let $f,g \in I$.…

## 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 a subring is an ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.9 Solution: We have already seen which of these are subrings. (1) Let \$S = \{ f…

## Decide whether a given subset is a subring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.6 Decide which of the following are subrings of the ring of all functions from the closed…