Michael Reid <reid@math.ucf.edu> wrote:
Daniel Asimov wrote:
I don't think I saw any mention of it here, but Kaplansky passed away little over one month ago, on June 25. Here's a link to a nice obituary . . . but I wasn't able to extract specific results of his from this:
<http://www.signonsandiego.com/uniontrib/20060714/news_1m14kaplansk.html>
Actually, I'm curious -- if any cognoscenti would care to explain some theorems he's famous for, I'd be very interested.
i'm afraid i can't really do it justice, but there hasn't been much response, so i thought i'd try.
one result of his i know is that a projective module over a (non-commumtative) local ring is free. it's possible that the finitely generated version predates him, and the general case is his contribution.
One can gain some idea of Kaplansky's work and its influence by searching books.google.com for "Kaplansky theorem". This yields 344 hits - more than most of his contemporaries. See the list below which includes a random sample of his contemporaries along with a mix of older luminaries for comparison. Of course one shouldn't take such a list too seriously.
also, his book "infinite abelian groups" is well-known. (oops, i see that's mentioned in the obit.)
He was a very gifted expositor. For many years I've kept his "Commutative Rings" always within reach (now on my laptop). Cauchy 5840 Gauss 5590 Weierstrass 5290 Euler 5230 Noether 3700 Lagrange 3160 Fourier 2150 Borel 1830 Weyl 1710 Godel 1530 Birkhoff 1460 Neumann 1440 Weil 1440 Witt 1100 Hilbert 842 Riemann 805 Jacobi 789 Hall 770 Laplace 709 Kronecker 625 Serre 621 Grothendieck 468 Newton 454 Hardy 411 Zassenahus 362 Kaplansky 344 **** Dilworth 320 Dedekind 295 Thom 259 Kac 255 Milnor 233 Nash 231 Quillen 229 Turing 204 Deligne 186 Faltings 177 Poincare 156 Wiles 117 Donaldson 115 Thurston 115 Erdos 101 --Bill Dubuque