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18.706 HOMEWORK 5

By Due Wed and May In Class

Abstract

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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