# Celebratio Mathematica

## Irving Kaplansky

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.