The G-Function of Macrae

Abstract

Let R be a commutative ring with identity. A finitely generated R-module M is called a torsion module if the annihilator of M contains a non zero-divisor. In [18] MacRae proved the followingTheorem. If R is a noetherian ring then there is a map G with the following properties from the class of torsion R-modules of finite homological dimension to the set of integral invertible ideals of R.(i) If M is a finitely generated torsion R-module with homological dimension ≦ 1 then G(M) = F(M), the first Fitting ideal of M.(ii) If S is a multiplicative subset of R then G(Ms) = G(M) s.(iii) If 0 → L → M → N → 0 is an exact sequence of torsion modules of finite homological dimension then G(M) = G(L)G(N).