Let R be a commutative ring with multiplicative identity 1 0. Call an ideal I in R prime if for any x, y Є R with xy Є I, we must have either x = I or y Є I. Prove that if I is not prime, then there exists an ideal J of R with I ÇJ Ç R. [Hint: Saying I is not prime means there exist x, y Є R such that xy Є I but x, y ‡ I. Now set J=I+xR and prove that I ‡ J and J ‡ R.]

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