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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) dx \\=&\ \int_0^1 f(x) dx + \int_0^1 g(x) dx \\=&\ \varphi(f) + \varphi(g). \end{align*}Thus $\varphi$ is an additive group homomorphism.

On the other hand, note that $\varphi(x^2) = 1/3$, while $\varphi(x)^2 = 1/4$. Thus $\varphi$ is not a ring homomorphism.


Linearity

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