### Math 818: Problem set 9, due April 1 (no fooling!)

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, April 1, 2005. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)

Assume all rings are commutative with 1 not equal to 0 unless specifically stated otherwise.
1. Make a list of abelian groups of order 720 such that every abelian group of order 720 is isomorphic to a group on your list, but no two groups on your list are isomorphic.
2. Let n be an integer bigger than 1. Let R be the ring Z/(n).
• (a) Prove that an R-module is the same thing as an abelian group A such that na = 0 for each element a of A.
• (b) For any ring S, a cyclic S-module is a module of the form S/I, where I is an ideal. Using the fact that R-modules are particular abelian groups, state a structure theorem for R-modules that says that every finitely generated R-module is a product of cyclic R-modules of a particular form (what is that form?) in a unique way.
3. Problem 1 on p. 486.
4. Problem 3 on p. 486.
5. Problem 10 on p. 491.