THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

Abstract

Abstract. In this paper, we introduce and study the w-weak globaldimension w-w.gl.dim(R) of a commutative ring R. As an application, itis shown that an integral domain R is a Pru¨fer v-multiplication domainif and only if w-w.gl.dim(R) 6 1. We also show that there is a largeclass of domains in which Hilbert’s syzygy Theorem for the w-weak globaldimension does not hold. Namely, we prove that if Ris an integral domain(but not a field) for which the polynomial ring R[x] is w-coherent, thenw-w.gl.dim(R[x]) = w-w.gl.dim(R). 1. IntroductionRecall that a Pru¨fer domain is an integral domain in which every finitelygenerated ideal is invertible. It is well-known that the concept of Pru¨fer do-mains has played a central role in the development of the classical ideal theory.An important generalization of the Pru¨fer domain notion is that of a Pru¨ferv-multiplication domain (PVMD). This notion comes from multiplicative idealtheory, and various ideal-theoretic properties of it have been considered bymany authors, see for example [1, 2, 4, 7, 9, 10, 13, 14, 15]. From the homolog-ical algebra point of view, Pru¨fer domains are exactly the integral domains ofweak global dimension at most one. The original motivation for this work is toprovide a homological algebra characterization of PVMDs. To do so, we needthe notion of a w-flat module. Recently, modules of this type have receivedattention in several papers in the literature, see for example [3, 11, 21, 22].Throughout, R denotes a commutative ring with an identity element and allmodules are unitary.Now, we review some definitions and notation. Let J be an ideal of R.Following [25], J is called a Glaz-Vasconcelos ideal (a GV-ideal for short) if J isfinitely generated and the natural homomorphism ϕ : R → J