## Ring Theory Exam ## 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.