Let R be a commutative ring and let SCR is closed under multiplication. Let a to...

 

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Let R be a commutative ring and let SCR is closed under multiplication. Let a to be an ideal in R such that anS=0. We further assume that if b is an ideal in R such that a Cb and a b, then bns‡0. (a) Prove that a is a prime ideal in R. (b) Prove that if a is an ideal satisfying the above condition for S = {1}, then a is a maximal ideal. (c) Prove that if m is a maximal ideal in R, then m satisfyies the above condition for S = {1}. Let R be a commutative ring and let SCR is closed under multiplication. Let a to be an ideal in R such that anS=0. We further assume that if b is an ideal in R such that a Cb and a b, then bns‡0. (a) Prove that a is a prime ideal in R. (b) Prove that if a is an ideal satisfying the above condition for S = {1}, then a is a maximal ideal. (c) Prove that if m is a maximal ideal in R, then m satisfyies the above condition for S = {1}.

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a To show that a is a prime ideal we need to show that for any elements x y in R if xy is in a then ... View the full answer