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# st21 - S-1 R if and only if S ∩ M =/0(17 points 3 Let R...

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Math 5125 Monday, October 24 Sample Second Test no. 1. Answer All Problems. Please Give Explanations For Your Answers. 1. Let R be a commutative ring and let I denote the set of nilpotent elements of R (that is, { r R | r n = 0 for some positive integer n } ). Prove that I ± R and that R / I has no nonzero nilpotent elements. (16 points) 2. Let R be a commutative ring, let M be a maximal ideal of R , and let S be a multiplica- tively closed subset of R which contains 1. Prove that S - 1 M is a maximal ideal of
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Unformatted text preview: S-1 R if and only if S ∩ M = /0. (17 points) 3. Let R be a UFD. Prove that R [ x ] has inﬁnitely many irreducible elements, no two of which are associates. (17 points) Test on Wednesday, November 2. Material Chapters 7,8,9 approximately. One of the problems will be identical to one of the ungraded homework problems and one of the problems will be identical to one of the sample test problems....
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