Celebratio Mathematica

Irving Kaplansky

Kap: a tale of two cities

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.


[1]1989 Steele Prizes awar­ded at Sum­mer Meet­ing in Boulder,” No­tices Amer. Math. Soc. 36 : 7 (1989), pp. 831–​836. MR 1010381

[2]I. Ka­plansky: Se­lec­ted pa­pers and oth­er writ­ings. Spring­er (New York), 1995. With an in­tro­duc­tion by Hy­man Bass. MR 1340874 Zbl 0826.​01039

[3]I. Ka­plansky: “A sa­lute to Euler and Dick­son on the oc­ca­sion of Hy’s 65th birth­day,” pp. 79–​81 in Al­gebra, \( K \)-the­ory, groups, and edu­ca­tion (Columbia Uni­versity, New York, 1997). Edi­ted by T.-Y. Lam and A. R. Ma­gid. Con­tem­por­ary Math­em­at­ics 243. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1999. Con­fer­ence on the oc­ca­sion of Hy­man Bass’s 65th Birth­day. MR 1732041 Zbl 1017.​11014