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Irving Kaplansky Selected Papers and Other Writings

Irving Kaplansky
Publication Date: 
Number of Pages: 
Springer Collected Works in Mathematics
[Reviewed by
Fernando Q. Gouvêa
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Springer is engaged in the project of reprinting, in (relatively) affordable paperback editions, the collected/selected works of mathematicians they originally published in dark blue hardcover editions. The new editions are part of a new series, Springer Collected Works in Mathematics.

Kaplansky’s Selected Papers and Other Writings was originally published in 1995. The selection was presumably made by Kaplansky, who also adds “afterthoughts” to each of the papers. The “other writings” in the title are eight short unpublished pieces for which “Kap” must have had special affection.

Given Kaplansky’s standing and reputation among twentieth century mathematicians, the most surprising thing about the book is its size. A complete bibliography of Kaplansky’s work until 1995 is included, and it lists 126 items. By my count, eight of those are books, so the 22 papers selected for this book represent less than 20% of his publications. That suggests that the criteria for selection were rather draconian!

As one would expect, many of the papers here deal with rings. What surprised me was the large number of cases where these rings have topological structure: fields with valuations, rings of continuous functions, Banach algebras, operator algebras. I had not realized that Kaplansky had done so much in that area. Of course, many other topics show up, from combinatorics and game theory early on to homological algebra and Lie theory later on.

Kaplansky’s “afterthought” sections are fascinating. He is very candid. Sometimes he explains what got him interested in the topic. Other times he will offer corrections or extensions. He points to other work, especially when it completes the program he initiated. One “afterthought” opens with “I imagine that for some time no one has read this paper for its mathematical content, because it has been largely superseded.” It appears that the reason it is included is so that Kaplansky can offer “a better proof and an admission of error.”

I am a fan of the Collected Works in Mathematics series in general, but I’d say that this volume is especially attractive. Kaplansky writes very well, and as a result many of these papers are easy to read. A few might even be readable by undergraduates and could serve as the seed of interesting research projects. And several of the papers are classics.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He is the editor of MAA Reviews.

Preface.- Bibliography of the Publications of Irving Kaplansky.- Introduction by Hyman Bass.- 22 Original Papers.- 8 Other Writings.- Permissions.