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Unformatted text preview: R . 4. Let R be an integral domain and let Q be its eld 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 eld k . Prove that every nitely generated k [ x,y ] is noetherian. 1...
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This note was uploaded on 06/19/2011 for the course MATH 601 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
- Spring '11