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 Rmodule? (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 Rmodule is projective. (b) Every Rmodule 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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 Spring '11
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 Algebra, Ring, Ring theory, Commutative ring, prime ideal

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