My research speciality is the theory of finite rank torsion free abelian groups. (However for the past fifteen years most of my results have been stated in the context of torsion free modules over dedekind domains, since most results in abelian group theory generalize to that realm with the aid of only minor adjustments.) Although during the past twenty years there have been important contributions to the field from a number of notable mathematicians, I believe that I deserve much of the credit for determining the overall shape of research in this area.
Abelian groups would seem to be among the very simplest of algebraic structures. They have only a single algebraic operation, usually denoted as addition. The fact that this operation is commutative trivializes the bulk of the concepts of traditional group theory. The absence of torsion eliminates another major source of complexity, and the requirement of finite rank restricts the difficulty still further. In fact, finite rank torsion free abelian groups can be characterized simply as the subgroups of finite dimensional vector spaces over the field Q of rational numbers. Furthermore it is well known that finitely generated torsion free abelian groups (or modules over principal ideal domains) are free and so have essentially trivial structure.
Nonetheless, the study of finite rank torsion free abelian groups has been traditionally considered very difficult. In the book by Fuchs on abelian group theory which came out in 1973 (a year after I got my Ph.D.), only six sections out of 129 total are devoted to the finite rank torsion free case. (An unpublished result of mine is mentioned in a footnote!) A few very good mathematicians (Reinhold Baer, Richard S. Pierce, Bjarni Jónsson, M.C.R. Butler, Joseph Rotman, and Fuchs, among others) had written one or two important papers in the field and then moved on to other more fruitful areas. My teacher, David Arnold and I were almost the first abelian group theorists of any stature to devote themselves primarily to the finite rank torsion free realm.
At the time I got my degree, what most people knew about finite rank torsion free groups was various examples of pathological direct sum decompositions. It was known, for instance, that for any positive integer n there exist groups of rank 3 having n different non-isomorphic direct sum decompositions. But in 1973, I proved that there do not exist finite rank groups with infinitely many non-isomorphic decompositions. Furthermore, David Arnold and I wrote a joint paper in which we gave numerous conditions which ensure that direct sum decompositions are well behaved, thus indicating that ill-behaved decompositions are in fact exceptional. (This line of thought was later followed up very effectively by Robert Warfield.) In 1973, I also developed the key concept of near isomorphism for groups and showed that the ideas of Jacobinski, Reiner, and others from the theory of lattices over orders could be adapted to give new insights into the structure of finite rank groups.
Later on (about the time I first became affiliated with the University of Hawaii) I introduced the notion of the splitting field of a finite rank group and showed how this related to a class of groups originally studied by M.C.R. Butler in 1965. By introducing the term "Butler groups," I drew attention to this important class of groups, which has subsequently become one of the major foci of attention in the theory. At the same time, I showed how the striking results in the representation theory of finite dimensional algebras which had been recently been obtained by mathematicians like Gabriel, Dlab, Ringel, Auslander, and Reiten could be adapted to provide key insight into the most fundamental of all questions in the field: To what extent can finite rank torsion free abelian groups be classified? (The answer, not surprisingly, was that classification theorems can be expected only for very restrictive special classes of groups.)
Unfortunately, the Gabriel-Dlab-Ringel representation theory was not at that time very accessible to abelian group theorists and I started frequently having mathematicians tell me that they could see that what I was doing was very deep but that they personally found the concepts I was using much too difficult. (Richard Pierce wrote in Math Reviews in 1982, "The author has promised to write a self-contained leisurely exposition of his work on Butler modules. Such a monograph would be gratefully received by many of us mortals.")
In 1983, I tried to remedy this by writing up a set of notes ("A Seminar on Splitting Rings for Torsion Free Modules over Dedekind Domains") published as part of the proceedings of the conference on abelian group theory hosted by myself and Adolf Mader here in Honolulu in the winter of 1982-83. These notes laid the fundamental concepts out from the beginning out as simply as possible and were meant to have almost no prerequisites except basic algebra and abelian group theory. This "seminar" was partly successful, although I was somewhat discouraged to find that most abelian group theorists still found working through it to be a major chore. (Pierce's comment in Math Reviews (1985): "The paper under review is an elegant, lucid survey of a mature area in the theory of modules over Dedekind domains; it will greatly smooth the path to a full appreciation of the author's contribution to this subject.")
More recently, in my 1989 paper on Warfield Duality, I focused attention on the class of locally free groups. In my opinion, the theory of finite rank torsion free abelian groups can be seen as a spectrum with quotient divisible groups (essentially the class dealt with in my splitting field papers) at one end of it and locally free groups at the other. And yet the class of locally free groups had been totally neglected from the time of Warfield's fundamental paper in 1968 (whose results are much more far reaching than most people had realized) until my own in 1989.
I have certainly proved my share of good theorems in my lifetime. However there has been a subtext to most of my papers which I considered more important than the new results they presented. Namely, my hidden objective has been to transform the landscape of torsion free abelian group theory by introducing new concepts and language and changing the way people think about the subject. A small part of this agenda has been my effort to get people to see the theory of finite rank torsion free abelian groups as a subspecialty of commutative ring theory rather than of group theory. To this end, I have stated my results in the context of modules over dedekind domains and as much as possible replaced specialized terminology from abelian group theory by the generally accepted terminology of ring theory -- saying "finite length" rather than "finite," "non-faithful" instead of "bounded," and "essential submodule" instead of "full subgroup."
In addition to presenting new results, my papers often took key work from the past and presented it anew in what seemed to me a more coherent, natural form. I have already mentioned my efforts in getting the class of groups defined by M.C.R. Butler in 1965 the attention it deserved, and in enabling people to see Warfield's 1968 paper on locally free groups in its true generality. As part of my work on splitting fields, I took a duality functor which David Arnold had constructed in a rather clumsy computational fashion in his dissertation (published in 1972) and redefined it in a new conceptual way that showed how it related to the classic notion of duality for finite dimensional vector spaces. I christened this construction "Arnold Duality" and later showed how it could be used to solve the problem of finding the maximal divisible subgroup of the tensor product of two groups.
As part of my "Seminar," I took the concept of the field of definition for a finite rank torsion free ring, which Beaumont and Pierce had presented in segments scattered over three different papers, and gave a simple unified presentation for it, relating it to the the notion of integral closure in commutative ring theory.
Trying to present a new framework for an area of mathematics by smuggling it into papers presenting specialized results is probably not a very sensible approach, however, and I can't say that I was very successful in this respect. So when I sat down in 1989 to write a proposal for an upcoming sabbatical, I finally realized something rather obvious (although the thought was certainly not completely welcome): I realized that the time had come for me to write a book. Not only would this be the only reasonable way for me to really present my way of thinking, but there is an obvious need for such a book. The material on finite rank torsion free groups in Fuchs's book (1973) is completely outdated now and the only other exposition in book form is a set of notes by David Arnold (1982) which is in very rough form and in many places barely readable. Besides, writing a book is what academics on sabbatical traditionally do, isn't it?
After about two years of very hard work, on this book, though, I realized that I had bitten off more than I cared to chew. And various things started happening in my personal life which caused me to direct my energy in very different directions. In fact, about a year after I returned from my sabbatical I reached the point where I was no longer able to stand looking at the damn book, even though it must be at least 95% finished.
Finally, though, the World Wide Web has given me an opportunity to offer it to the world, albeit in unfinished form. The information is available to anyone who wants it, and that's all anyone needs.
Footnote: Why I Stopped Doing Mathematical Research.