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Ring homomorphisms preserve nilpotency

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.32

Solution: Suppose $x^n = 0$. Then $$\varphi(x)^n = \varphi(x^n) = \varphi(0) = 0,$$ so that $\varphi(x)$ is nilpotent.


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