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2008 Jan

2008 January

  1. State a structure theorem for finitely generated modules over a PID, including uniqueness conditions on the direct summands.

  2. Suppose that \(f: M \rightarrow N\) and \(g : A \rightarrow B\) are homomorphisms of right and left \(R\)-modules, respectively. Prove that there is a group homomorphism \(h: M \otimes_R A \rightarrow N \otimes B\) with \(h(m \otimes a) = f(m) \otimes g(a)\) for all \(a\in A\) and \(m\in M\).

  3. Let \(A\) be an \((R,S)\)-bimodule and \(B\) be an \((R,T)\)-bimodule and let \(M = \operatorname{Hom}_R(A,B)\).\ a. Give the actions of \(S\) and \(T\) on \(M\) making it an \(S\)-module and a \(T\)-module (left or right, as appropriate). No proofs required.\ b. Assuming your module actions from the first part, prove that \(M\) is in fact an \((S,T)\) or \((T,S)\) bimodule (whichever is appropriate).

  4. Give an example of each of the following or prove that none exists (no proof required).\ a. An injective \(\mathbb Z[i, \pi]\)-module. (You may assume that \(\pi\) is transcendental over \(\mathbb Q\).)\ b. A ring which is a UFD but which does not have the ascending chain condition on ideals.\ c. Two nonzero \(\mathbb Q\)-modules \(M\) and \(N\) with \(M\otimes_{\mathbb Q} N = 0\).\ d. A Euclidean domain which is not a PID.