Introduction

It is well known that, for a Gorenstein ring A, the total ring of fractions of A provides the injective envelope of A. One of the motivations behind the work which led to our construction of modules of generalized fractions (which was outlined in a lecture at the Symposium, is reviewed in §2 below, and is described in detail in [9]) was a desire to find a similarly satisfactory description of the terms Ei(A) for i > O in the minimal injective resolution for the Gorenstein ring A.

At the end of this paper it is shown that generalized fractions do provide such descriptions: whenever R is a commutative ring (with identity), M is an R–module, n is a positive integer and U is what is called a triangular subset of Rn, a module U−nM of generalized fractions may be constructed; in the case of a Gorenstein ring A, the set Un of all poor A–sequences of length n forms a triangular subset of An (we say that a sequence a1, …, an of elements of A forms a poor A–sequence if

((Aa1 + … + Aai−1) : ai) = (Aa1 + … + Aai−1)

for all i 1, …, n), and the module of generalized fractions U−nnA turns out to be isomorphic to En−1 (A).