Additive rank functions in Noetherian rings (Q910471)

An additive rank function on a ring R is a function that assigns to each finitely generated R-module M a non-negative integer \(\lambda\) (M) such that \(\lambda (M)=\lambda (N)+\lambda (L)\) for each short exact sequence \(0\to N\to M\to L\to 0\). An important example is Goldie's reduced rank function. It is obvious that any non-negative integer linear combination of additive rank functions is again an additive rank function and so a natural question is to find the extreme or atomic rank functions. This is done in the present paper for the case of a right noetherian ring: there are a finite number of atomic rank functions, one corresponding to each minimal prime ideal of R. The construction of such an atomic rank function \(\rho\) corresponding to the minimal prime P of R is similar to that of the reduced rank: if N is the nilpotent radical of R then \(\rho (M)=\sum \rho (MN^ i/MN^{i+1})\), while if \(MN=0\) then \(\rho (M)=length(M\otimes Q(R/P))\), where Q(R/P) is the simple artinian quotient ring of R/P. An arbitrary additive rank function \(\lambda\) can then be expressed as \(\lambda =\sum k_ i\rho_ i\), where \(\rho_ i\) are the atomic rank functions and each \(k_ i\) is a non-negative integer. Two examples are presented to illustrate the usefulness of this decomposition in calculations.