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## Main Question or Discussion Point

Let R be a ring with ideals I, J, and P. Prove that if P is a prime ideal and I intersect J is a subset of P, then I is a subset of P or J is a subset of P.

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Let R be a ring with ideals I, J, and P. Prove that if P is a prime ideal and I intersect J is a subset of P, then I is a subset of P or J is a subset of P.

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- #3

disregardthat

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If neither I nor J is contained in P, some element a in I and b in J are neither in P, as you say. But this is a contradiction, since ab is in P and P is prime, hence one of the ideals is contained in P.

Follow-up: Show this for an arbitrary finite number of ideals.

Follow-up: Show this for an arbitrary finite number of ideals.

Last edited:

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Deveno

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but since I is an ideal, ab is in I. since J is an ideal ab is in J. therefore ab is in I∩J, and thus in P. since P is prime, either a is in P, contradicting our choice of a, or b is in P, contradicting our choice of b.

the only conclusion is that we cannot pick such a in I AND b in J outside of P, either I or J must lie within P.

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disregardthat

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