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AlgebraQual2008aug - R 4 Let R be an integral domain and...

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August 2008 Qualifying Examination Algebra Part There are only 6 questions. Do them all. 1. Let A be a finite abelian group. Prove that A is not a projective Z -module and also that it is not an injective Z -module. 2. Prove that Q / Z Z Q / Z = 0 3. Let I be an ideal of a commutative ring R and let the radical of I be defined as I = { r R | r n I for some n 0 } (a) Prove that I is an ideal of R containing I , and that if I is a prime ideal, then I = I . (b) An ideal Q of R is called primary if whenever ab Q and a / Q , then b n Q for some n 1. Prove that the primary ideals of Z are 0 and ( p n ) where p is a prime number and n is a positive integer. Prove also that if Q is a primary ideal of a ring R , then Q is a prime ideal in
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Unformatted text preview: R . 4. Let R be an integral domain and let Q be its field of fractions. Prove that tensoring with Q over R is exact. Does Q have to be projective too as an R-module? (This could be a little tricky). Prove it, or give a counterexample. 5. Let R be an artinian ring. Prove that the following are equivalent: (a) Every R-module is projective. (b) Every R-module is injective. (c) R is a semisimple ring. 6. Let k [ x,y ] denote the polynomial ring in two variables over a field k . Prove that every finitely generated k [ x,y ] is noetherian. 1...
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