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Let P be a prime ideal in a commutative ring R and let S = R\P, i.e. the complement of P in R. Pick out the true statements: (a) S is closed under addition. (b) S is closed under multiplication. (c) S is closed under addition and multiplication

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-1 for using the imperative "pick out the true statements" in a question addressed to actual people, which contribute here just for fun. We're not obliged to answer you. Consider giving your question a friendlier tone (e.g. "Please help me picking out the true statements..."), and some background on what you've tried. – Nils Matthes Sep 18 '12 at 10:28
I don't care. I'm used to mathematical questions stated in the imperative. And I don't think I answer questions only for the questioners. There are several reasons why I answer(e.g. for fun, for reps, just bored, etc.). – Makoto Kato Sep 18 '12 at 10:38
Can you tell us your thoughts on the problem? – Alex Becker Sep 19 '12 at 19:11

(a) is false(e.g. $\mathbb{Z} - 2\mathbb{Z}$).

(b) is true.

(c) is false, since (a) is false.

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