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Please type or print clearly. Write the exercise number on your paper (but you don’t have to rewrite the question). Justify your assertions. Changes I made to the original exercise are in bold face. Exercises from [AM]: Assume the rings are noncommutative unless stated otherwise. 2.1 Show that (Z/mZ) ⊗Z (Z/nZ) = 0 if m, n are coprime. 2.3 Let A be a commutative local ring, M and N finitely generated A-modules. Prove that if M ⊗N = 0, then M = 0 or N = 0. [Hint: Let m be the maximal ideal, k = A/m the residue field. Let Mk = k ⊗A M ∼ = M/mM by class notes 4.6. By Nakayama’s lemma, Mk = 0 = ⇒ M = 0. But M ⊗A N = 0 = ⇒ (M ⊗A N)k = 0 = ⇒ Mk ⊗k Nk = 0 = ⇒ Mk = 0 or Nk = 0, since Mk, Nk are vector spaces over a field.] 2.5 Let A[x] be the ring of polynomials in one indeterminate over a ring A. Prove that A[x] is a flat A-bimodule. [Hint: Use Class notes 4.12] 2.6 For any right A-module M, let M[x] denote the set of all polynomials in

Year: 2011

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