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Auslander and Bridger have shown that, under conditions somewhat weaker than finite projective dimension, the "torsion freeness" properties of a module M (e.g. being reflexive, being the kth syzygy of another module) are determined by certain arithmetic conditions on the Exti(M, R). In this paper it is shown that a single ideal, the intersection of the annihilators of these modules, gives this same information. This ideal is then related to the Fitting invariants and invariant factors of M, and a computation is made of certain syzygies of a quotient of M (by a regular M-sequence). Introduction. In [2] it is shown that under certain reasonable hypotheses (somewhat weaker than having finite projective dimension), the torsion freeness properties of a module M are determined by the grades of the modules Exti(M, R) for i>0. These numbers, in turn, depend only on the ideals Wi=rad[Ann Exti(M, R)]. Denoting by y(M) the intersection of these ideals, the author has shown that y(M) likewise determines the torsion freeness properties of M. In this paper this result will be partially generalized, and an explicit calculation made exhibiting the torsion freeness properties (?3). In ?2 we relate y(M) to the Fitting invariants and invariant factors of M. Unless otherwise specified, all rings will be commutative and noetherian and modules finitely generated (hence finitely presented). 1. The ideals ,B(M) and y(M). Although M is projective if and only if Ext1(M, -)=0, one need not test Ext1(M, -) on all modules. Let us write QM=Q1M=ker(P-+M) where P-+M is any map of a projective module P onto M. If we agree to call two objects A and B projectively equivalent when A +P-B+Q for projective objects P and Q, then it is well known that the projective "equivalence class" of QM depends only on that of M (see Lemma 10 below). LEMMA 1. M is projective if and only if Ext1(M, Q2m)=0. Received by the editors February 5, 1971. AMS 1970 subject classifications. Primary 13C10, 13C15, 13DO5.

An ideal criterion for torsion freeness