#### by Susanna S. Epp and E. Graham Evans, Jr.

We are two of the fifty-five students who completed a doctorate with Kaplansky between 1950 and 1978. This is an astonishing number. Indeed during the years 1964–1969, when we were at the University of Chicago, Kap oversaw an average of three completed dissertations a year despite serving as department chair from 1962–1967. His secret, we think, was an extraordinary instinct for productive avenues of research coupled with a generous willingness to spend time working with his students. He also often encouraged students to run a seminar, with beginning students presenting background material and advanced students presenting parts of their theses.

When Evans worked with him, Kap was teaching the commutative algebra
course that was published soon afterward by
Allyn and Bacon. As with
each course he taught, he filled it with new thoughts about the
subject. For instance, at one memorable point he experimented to see
how much he could deduce if he knew only that __\( \operatorname{Ext}^1(A,B) \)__ was zero. He
managed to get pretty far, but eventually the proofs became
unpleasantly convoluted. So he abruptly announced that henceforth, he
would assume the full structure of __\( \operatorname{Ext}^j(A,B) \)__, and the next
day he resumed lecturing in his usual polished fashion. This episode
was atypical in that he first developed and then cut off a line of
inquiry. More frequently, after commenting on new insights of his own,
he would interject questions for students to explore and develop. In
his lectures he made the role of non-zero divisors, and hence regular
sequences, central in the study of commutative rings. At one point he
gave an elegant proof, avoiding the usual filtration argument, that
the zero divisors are a finite union of prime ideals in the case of
finitely generated modules over a Noetherian ring. Then he asked Evans
to try to determine what kinds of non-Noetherian rings would have the
property that the zero divisors of finitely generated modules would
always be a finite union of primes. One of the ideas in Kap’s proof
was just what Evans needed to get the work on his thesis started.

The year that Epp worked with Kap, he was not teaching a course but had gone back to a previous and recurring interest in quadratic forms. A quintessential algebraist, he was interested in exploring and expanding classical results into more abstract settings. Just as in his courses he tossed out questions for further investigation, in private sessions with his students he suggested various lines of inquiry beyond his own work. In Epp’s case this meant exploring the results Kap had obtained in generalizing and extending H. Brandt’s work on composition of quaternary quadratic forms and trying to determine how many of these results could be extended to general Cayley algebras.

Kap typically scheduled an early morning weekly meeting with each student under his direction. For some it was much earlier than they would have preferred, but for him it followed a daily swim. He led our efforts mostly by expressing lively interest in what we had discovered since the week before and following up with question after question. Can you prove a simpler case? Or a more general one? Can you find a counterexample? When one of us arrived disappointed one day, having discovered that a hoped-for conjecture was false, Kap said not to be discouraged, that in the search for truth negative results are as important as positive ones. He also counseled persistence in other ways, commenting that he himself had had papers rejected — a memorable statement because it seemed so improbable. Having made contributions in so many fields and having experienced the benefits of cross-fertilization, he advised being open to exploring new areas. Some of his students may have taken this advice further than he perhaps intended, ultimately working far from their original topic areas at the National Security Agency, at the Jet Propulsion Laboratory, and in K–12 mathematics education, for example.

Kap derived a great deal of pleasure from having generated 627 mathematical descendants, perhaps especially from meeting his mathematical grandchildren and great-grandchildren. When one encountered him at the MSRI bus stop one day and, not knowing what to say, commented on the weather, Kap responded with a smile, “Cut the crap. Let’s talk mathematics.” They did, and he became one of the many students Kap mentored long after he retired.