#### by T. Y. Lam

Through Professor Hyman Bass, Kap was my mathematical grandfather. This and the fact that Kap offered me my first job as an instructor at Chicago were perhaps not statistically independent events. The time was forty years ago, when Kap was finishing his five-year term as Chicago chair. The offer was consummated by a Western Union telegram — the 1960s equivalent of email. Kap didn’t ask for my C.V. (I wouldn’t have known what that was); nor did he want to know my “teaching philosophy” (I had none). For my annual salary, Kap offered me US\$8,000 — a princely sum compared to my then T.A. stipend of US\$2,000 at Columbia University. I have joked to my colleagues that I’ll always remember Kap as the only person through my whole career to have ever quadrupled my salary. But in truth, a ticket to Chicago’s famed Eckhart Hall for postdoctoral studies was more than anything a fledgling algebraist could have dreamed. For this wonderful postdoctoral experience Kap afforded me through his unconditional confidence in a mathematical grandson, I have always remained grateful.

I met Kap for the first time in the fall of 1967 when I reported to work in Hyde Park. By that time, Kap had already taught for twenty-two years at the University of Chicago. Although he was Canadian by birth, Chicago had long been his adopted home and workplace: it is, appropriately, the city where our “tale” begins.

For students interested in abstract algebra, Kaplansky is virtually a household name. In graduate school, I first learned with great delight Kap’s marvelous theorem on the decomposition of projective modules, and his surprisingly efficient treatment of homological dimensions, regular local rings, and UFDs. It was to take me forty more years, however, to get a fuller glimpse of the breadth and depth of Kap’s total mathematical output. In these days of increasing specializations in mathematics, we can only look back in awe to Kap’s trail-blazing work through an amazingly diverse array of research topics, ranging from valuation theory, topological algebra, continuous geometry, operator algebras and functional analysis, to modules and abelian groups, commutative and homological algebra, P.I. rings and general noncommutative rings, infinite-dimensional Lie algebras, Lie superalgebras (supersymmetries), as well as the theory of quadratic forms in both its algebraic and arithmetic flavors. Kap was master of them all. In between the “bigger” works, Kap’s publications also sparkled with an assortment of shorter but very elegant notes, in number theory, linear algebra, combinatorics, statistics, and game theory. All of this, still, did not include the many other works recorded in “fourteen loose-leaf notebooks” (referred to in the preface of [2]) that Kap had kept for himself over the years. One cannot help but wonder how many more mathematical gems have remained hidden in those unpublished notebooks!

For me, reading one of Kap’s papers has always proved to be a richly rewarding experience. There are no messy formulas or long-winded proofs; instead, the reader is treated to a smooth flow of novel mathematical ideas carefully crafted to perfection by an artisan’s hand. Some authors dazzled us with their technical brilliance; Kap won you over by the pure soundness of his mathematical thought. In his publications, Kap was much more given to building new conceptual and structural frameworks, than going down single-mindedly into a path of topical specialization. This style of doing mathematics made him a direct intellectual descendant of Emmy Noether and John von Neumann. As a consequence, many of Kap’s mathematical discoveries are of a fundamental nature and a broad appeal. The famous Kaplansky Density Theorem for unit balls and his important inaugural finiteness result in the theory of rings with polynomial identities are only two of the most outstanding examples.

Those of us who have had the privilege of listening to Kap all knew that he was extremely well spoken and had indeed a very special way with words. However, this gift did not always manifest itself when Kap was in social company with Chellie. It was quite clear to all his colleagues who Kap thought was the better orator in the family. Dinner parties the Kaps attended were often replete with Chellie’s amusing stories about the Chicago department and its many colorful mathematical personalities, from an austere André Weil down to the more transient, sometimes bungling graduate students over the years. As Chellie recounted such funny stories with her characteristic zest and candor, Kap would listen admiringly on the side — without interruption. Only at the end of a story would he sometimes add a clarifying comment, perhaps prompted by his innate sense of mathematical precision, such as “Oh, that was 1957 summer, not fall.”

Kap’s extraordinary gift in oral (and written) expression was to find
its perfect outlet in his teaching, in which it became Chellie’s turn
to play a supporting role. In the many lecture courses Kap gave at the
University of Chicago in a span of thirty-nine years, he introduced
generation after generation of students to higher algebra and
analysis. In those courses he taught that were of an experimental
nature, Kap often directly inspired his students to new avenues of
investigation, and even to original mathematical discoveries at an
early stage. (Schanuel’s Lemma on projective resolutions, proved by
Stephen Schanuel in Kap’s fall 1958 Chicago course in homological
algebra, was perhaps the best known example.) It was thus no accident
that Chicago graduate students flocked to Kap for theses supervision.
Over the years, Kap directed doctoral dissertations in almost every
one of the mathematical fields in which he himself had worked. Many of
Kap’s fifty-five Ph.D. students from Chicago are now on the senior
faculty at major universities in the U.S. Currently, the Mathematics
Genealogy Project listed Kap as having 627 descendants — and
counting. This is the *second* highest number of progeny produced by
mathematicians in the U.S. who had their own Ph.D. degrees awarded
after 1940. We leave it as an exercise for the reader to figure out
who took the top honor in that category, with the not-too-useful hint
that this mathematician was born a year after Kap.

While Kap had clearly exerted a tremendous influence on mathematics
through his own research work and that of his many Ph.D. students, the
books written by him were a class by themselves. The eleven books
listed in the sidebar on this page traversed the whole spectrum of
mathematical exposition, from the advanced to the elementary, reaching
down to the introduction of mathematics to non-majors in the
college. *Differential Algebra* typified Kap’s broad-mindedness in
book writing, as its subject matter was not in one of Kap’s specialty
fields. On the other hand, *Infinite Abelian Groups* introduced
countless readers to the simplicity and beauty of a subject dear to
Kap’s heart, while *Rings of Operators* served as a capstone for his
pioneering work on the use of algebraic methods in operator
algebras. *Lie Algebras*, *Commutative Rings*, as well as *Fields and
Rings*, all originating from Kap’s graduate courses, extended his
classroom teaching to the mathematical community at large, and
provided a staple for the education of many a graduate student
worldwide, at a time when few books covering the same materials at the
introductory research level were available. In these books, Kap
sometimes experimented with rather audacious approaches to his subject
matters. For instance, *Commutative Rings* will probably
go down on record as the only text in commutative
algebra that totally dispensed with any discussion
of primary ideals or artinian rings.

As much as his books are appreciated for their valuable and innovative
contents, Kap’s great fame as an author derived perhaps even more from
his very distinctive writing style. There is one common characteristic
of Kap’s books: they were all short — something like 200 pages
was the norm. (Even *Selected Papers*
[2]
had only 257
pages, by his own choice.) Kap wrote mostly in short and simple
sentences, but very clearly and with great precision. He never
belabored technical issues, and always kept the central ideas in the
forefront with an unerring didactic sense. The polished economy of
Kap’s writing makes it all at once fresh, crisp, and engaging for his
readers, while his mastery and insight shone on every page. The
occasional witty comments and asides in his books — a famous
Kaplansky trademark — are especially a constant source of
pleasure for all. In retrospect, Kap was not just a first-rate author;
he was truly a superb expositor and a foremost mathematical stylist of
his time.

After I moved from Chicago to Berkeley, my contacts with Kap became sadly rather infrequent. So imagine my great surprise and delight, sixteen years later, when word first came out that Kap was to retire from the University of Chicago, in order to succeed Chern as the director of MSRI! In the spring of 1984, the Kaplanskys arrived and established their new abode a few blocks north of the university campus — in Berkeley, California, the second city of our tale.

The math departments at Chicago and Berkeley share much more than the “U.C.” designation of the universities to which they belong. There has been a long (though never cantankerous) history of the Berkeley department recruiting its faculty from the Chicago community, starting many years ago with Kelley, Spanier, and Chern. Indeed, when Kap himself joined the U.C.B. faculty in 1984, there were at least as many as sixteen mathematicians there who had previously been, in one way or another, associated with the University of Chicago. It must have given Kap a tinge of “nostalgia” to be reunited, in such an unexpected way, with so many former graduate students, postdocs, and colleagues from his beloved Chicago department. But if anyone had speculated that, by coming West, Kap was to spend his golden years resting on his laurels, he or she could not have been more wrong. In fact, as soon as Kap arrived at Berkeley in 1984, he was to take on, unprecedentedly, two simultaneous tasks of herculean proportions: (a) to head a major mathematics research institute in the U.S., and (b) to preside over the largest mathematical society in the world — the AMS.1 Other contributors to this memorial article are in a much better position than I to comment on Kap’s accomplishments in (a) and (b) above, so I defer to them. In the following, my reminiscences on Kap’s Berkeley years are more of a personal nature. From 1984 on, I certainly had more occasions than ever before to interact mathematically with Kap — discussing with him issues in quadratic forms and ring theory. Kap seemed to favor the written mode of communication (over the oral), but his letters were just as concise as his books. I still have in my prized possession an almost comical sample of Kap’s terseness, in the form of a covering letter for some math notes he sent me. Written out on a standard-size 8 1/2 by 11 MSRI letterhead, the letter consisted of twelve words: “Dear Lam: I just did a strange piece of ring theory. Kap.” It was briefly — but of course unambiguously — dated: “Apr. 11 /97”.

Another interaction with Kap in 1998 led to some mathematical output.
In preparation for a special volume in honor of Bass’s sixty-fifth
birthday, I was very much hoping to commission an article from Kap. In
his usual self-effacing fashion, Kap protested that he had really
nothing to write about. However, after much persuasion on my part
(stressing that he must write for Bass), he gave in and wrote up in
his impeccable hand a short note in number theory
[3].
Glad that my tactics had paid off, I worked all night to set Kap’s
written note in ~~\TeX~~, and delivered a finished printout to him early
the next morning. Kap was surprised; he thanked me profusely, but said
that maybe he shouldn’t have written his article. It was too late.

One of Kap’s best known pieces of advice to young mathematicians was
to “spend some time every day learning something new that is disjoint
from the problem on which you are currently working, … and read the
masters”
[1].
Amazingly, even after reaching his seventies,
Kap still took his own advice personally and literally. In all the
years he was in Berkeley, Kap made it his habit to go to every
Monday’s Evans–MSRI talk and every Thursday’s Math Department
Colloquium talk. He even had a favorite seat on the left side of the
front row in the colloquium room, which, in deference to him, no local
Berkeley folks would try to occupy. In the years 1995–97 when I
worked at MSRI, I saw Kap quite frequently at the periodicals table in
the library, poring over the *Mathematical Reviews* to keep himself
abreast with the latest developments in mathematics. And he read the
masters too, e.g., in connection with his work on the integral theory
of quadratic forms. Members of MSRI have reported sightings of Kap
using a small step-ladder in the library to reach a certain big book
on a high shelf, and putting the book back in the same fashion after
using it (instead of leaving it stray on a table). That tome was an
English translation of *Disquisitiones Arithmeticae*: the fact that even
a six-foot-tall Kap needed a step-ladder to access it was perhaps
still symbolic of the lofty position of the work of the
twenty-year-old
Carl Friedrich Gauss.
My two-years’ stay at MSRI was rich with other remembrances about Kap.
Undoubtedly, a highlight was Kap’s eightieth birthday fest in March
1997, which was attended by three MSRI directors and six MSRI deputy
directors, as well as visiting dignitaries such as
Saunders Mac Lane, Tom Lehrer, and
Constance Reid. Another most memorable gathering was
the holiday party in December 1996, where a relaxed and jovial Kap
sang some of his signature songs for us all, accompanying himself on
the piano in the MSRI atrium. His energetic, sometimes foot-stomping
performance really brought down the house! It saddens me so much to
think that, now that Kap is no longer with us, these heart-warming
events will never be repeated again.

Twenty years may have been only about a third of Kap’s professional life, but I hope that Kap cherished his twenty years in Berkeley with as much fondness as he had cherished his thirty-nine years in Chicago. Those were the two cities (and universities) of his choice, for a long and very distinguished career in mathematics. In Chicago, Kap was a researcher, a chairman, a teacher, a mentor, and an author. In Berkeley, while remaining a steadfast researcher, Kap also became a scientific leader, a senior statesman, and a universal role model. In each of these roles, Kap served his profession with devotion, vigor, wisdom, and unsurpassed insight. His lifetime work has profoundly impacted twentieth century mathematics, and constituted for us an amazingly rich legacy. On a personal level, Kap — mathematical grandpa and algebraist par excellence — will continue to occupy a special place in my heart. I shall miss his great generosity and easy grace, but thinking of Kap and his towering achievements will always enable me to approach the subject of mathematics with hope and joy.