## The complete homomorphic preimage of a prime ideal is a prime ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.13 Solution: (1) By Exercise 7.3.24, $\varphi^\ast[P]$ is an ideal of $R$. Now suppose $ab \in \varphi^\ast[P]$.…