Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.10
Solution: Let $a,b \in R$ such that $ab = 0$. Since $P$ is a prime ideal and $ab \in P$, without loss of generality $a \in P$. If $a \neq 0$, then since $P$ contains no zero divisors in $R$, $b = 0$.
