----------------------------------------------------------------------- Algebra 2 Takehome Final ----------------------------------------------------------------------- Due Wednesday May 2, 2012 Construct an example of a pair (R,P) such that R is an integral domain and P is a projective R-module but not a free R-module. Then write a short expository paper about your construction, including a proof that it satisfies the requirement. (I hope that this can be done in 3-4 pages, but there are no requirements on the length.) You are welcome to speak to others or consult the literature to solve the problem, but your paper should be written individually. In general, paper has (at least) the following components: 1) A title, a list of authors, and a date. 2) An introduction. 3) The body of the paper. 4) A list of references. The introduction has two main goals: a) Make your readers interested in the subject, and b) briefly explain the history of the subject and how your new paper fits in there. For a) you might explain the "big picture", state main results, mention applications and relationships with other parts of mathematics. And of course, if your paper introduces new and revolutionary results or methods, then explain the great benefits to humanity resulting from this! Part b) includes mentioning earlier work on the same subject. In particular, if parts of your results, methods, or proofs etc. have appeared earlier, then point this out, stating where it can be found. A good and interesting account of the history can also greatly amplify the reader's interest. The body of the paper does the actual work, state definitions, prove theorems, give examples, etc. How all of this applies to your expository paper on projective modules is up to you. Anders