Question: Let R be a commutative ring with multiplicative identity 1 # 0. Call an ideal I in R prime if for any x,y R with

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

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