Matrices over polynomial rings
David Lissner · Transactions of the American Mathematical Society · 1961
DAVID LISSNER(') 1. Introduction.The polynomial rings of the title are the rings of polynomials in a finite number of variables with coefficients in a field.In a paper [2] published in 1955, J.-P.Serre asked the question, is every finitely generated projective module over such a ring free?It is an easy exercise to show that this is the case for polynomials in one variable; in a recent article [3] C. S. Seshadri has shown that it is also true for polynomials in two variables.Otherwise the question remains open today, and it is perhaps à propos to remark that Stephen Chase of the University of Chicago has shown that the statement (that every finitely generated projective module is free) may be true for a ring 7? and still fail to be true for RThe problem is equivalent to the following theorem: Let 7? be a polynomial ring, «i, • • ■ , anER, and (ai, • • • , an) = (1).Then 3 an wXw matrix M, with entries in R, first row (öi • • • c"), and | M\ ( = det M) = 1.(A proof of this equivalence will be given in §11.This proof has been known to Serre, and also to Kaplansky, for some time; it was communicated orally to the author by Professor Kaplansky.We include it here for completeness since it has not previously appeared in print.)The purpose of this paper is to study a special case of this matrix theorem, and a few similar theorems closed related to it.Specifically, we will attempt to determine for which polynomial rings 7? the following theorems hold.Theorem A. If 1G(oi.a2, a3) then 3 a 3X3 matrix M over R with first row (ai a2 a3) and | M\ = 1.Theorem B. If dE(ai, a2, a3), where d is any element of R, then 3 a 3X3 matrix M over R with first row (ai a2 a3) and \ M\ =d.Theorem C. If \E(a>i, a2, a3), then 3 2X2 matrices A and B over R such that (ai a2\ ) = AB -BA.