A seminar on splitting rings for torsion free modules over dedekind domains

in ``Abelian Group Theory,'' Lecture Notes in Mathematics 1006 (1983), pp.1 - 48.


Over the Christmas break in 1982-83, a conference on abelian group theory was held here in Honolulu at the East-West Center.

This conference was Adolf Mader's idea. It was about the last thing I wanted at the time. since I had been going through a period of rather profound depression, in part related to my financial situation, which was then so dismal that I felt that there would soon be no choice for me but to find a non-academic job, probably in the aerospace industry.

The last thing I wanted was to have a very large bunch of abelian group theorists showing up on my doorstep (as it were). But Adolf got me to agree to the conference by promising that he would do all of the work involved, which he in fact did. And when it actually took place, it turned out to be quite a bit of fun and a very worthwhile conference.

I decided that I wanted to write up a set of notes for a seminar I had presented two or three times here, while R.S. Pierce, Charles Vinsonhaler, Ray Mines, and David Arnold had been visiting. (Not all simultaneously.) Since I was formally one of the editors of the conference proceedings, I figured that if I submitted a paper it could scarcely be rejected, even if it was primarily expository, containing a minumum of new results. (By the time I finished writing the notes up, though, it turned out that there were more new results than I had expected. Furthermore, the proofs were mostly new and mostly much improved from those previously published by myself and others.)

At the time, I assumed that this would probably be the last mathematical paper I would ever write, and I put a lot of work into making it really complete and readable.

The first three paragraphs of the introduction to the paper are as follows.


This seminar is an introduction to the concepts deal with in [10] through [15]. It is in some sense a ``prequel'' to those papers, since it provides most of the background material needed to read them. It is based on extensive talks given in a disjointed, disorganized fashion to varying audiences in Honolulu from time to time during the past few years.

In writing papers, one feels that one has fulfilled one's obligation to the reader with respect to pre-existing mathematics if one provides precise and accurate citations to the literature. In talking before a live audience, unfortunately, one finds that this procedure is not tolerated. In the course of figuring out how to present this material in a self-contained way, I was struck by how simple much of the background material ultimately turned out to be. I was also struck by how much simpler the proofs I was presenting for my own theorems were than those in my published papers. Furthermore, I became aware that various things which in my papers I had either taken for granted or dismissed with a bit of hand waving were met with considerable puzzlement and somes skepticism from live audiences. And in some cases I found myself hard pressed to come up with careful, convincing proofs. In a few cases I found that my assertions were not exactly true.

Since it takes the better part of a semester to deliver this Seminar (at quite rapid pace), I eventually decided that it would turn out in the end to be less effort to write the whole thing down, rather than to continue giving it live whenever another visitor arrived for an extended stay in Honolulu. (Besides, eventually the permanent faculty here would get tired of hearing it.)