A note on generating sets for invertible ideals

Abstract

It is well known that an invertible fractional ideal of a commutative ring with identity must be finitely generated. S. U. Chase has shown that for any positive integer n, there is an integral domain Wn containing an invertible ideal with a basis of n, but no fewer, generators. Chase's example does not appear in the literature, but the example has been referred to by Bass in [1, 541], by Swan in [4, 270], and by Gilmer and Heinzer in [2 ]. We know of no verification of the details of Chase's example independent of Swan's results in [4]; these results of Swan are quite involved. In this note, we give an example of a domain Dn with identity containing an invertible ideal An with a basis of n, but no fewer, generators. The domain Dn is related to, but not isomorphic to, Chase's domain Wn. However, our verification that An has no basis of fewer than n elements depends only upon Lemma 1, a result which should be of independent interest in itself. The proof of Lemma 1 requires essentially only the Borsuk-Ulam Theorem [3, 152]. We use E to denote the field of real numbers; n is a positive integer greater than one, and {Xi}In1 is a set of indeterminates over E. LEMMA 1. If fi, f,_fCeE[X1, * * , Xn], where each nonzero monomial of each fi has odd degree, and if