Decomposition and classification of length functions

We study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

On the relation between length functions and exact Sylvester rank functions

Inspired by the work of Crawley-Boevey on additive functions in locally finitely presented Grothendieck categories, we describe a natural way to extend a given exact Sylvester rank function on the category of finitely presented left modules over a given ring R, to the category of all left R-modules.