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If a nontrivial prime ideal contains no zero divisors, then the ring is an integral domain

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 $R$, $b = 0$.


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