In a commutative unital ring, if a prime ideal contains a product of ideals then it contains one of the factors

Let R be a commutative ring with 1 \neq 0. Let I,J,P \subseteq R be ideals, and let P be prime in R. Suppose further that IJ \subseteq P. Prove that either I \subseteq P or J \subseteq P.


Suppose I \not\subseteq P. Then there exists a \in I such that a \notin P. Now let b \in J. Since IJ \subseteq P, ab \in P. Since P is a prime ideal and a \notin P, we have b \in P. Since b was arbitrary in J, J \subseteq P.

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