Let me suggest you some references:

Northcott Reufel, "A generalization of the concept of length" (http://qjmath.oxfordjournals.org/content/16/4/297.full.pdf)

Vamos, "ADDITIVE FUNCTIONS AND DUALITY OVER NOETHERIAN RINGS" (http://qjmath.oxfordjournals.org/content/19/1/43.full.pdf)

Zanardo, "Multiplicative invariants and length functions over valuation domains" (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jca/1323364358)

If your interest goes in this direction I know that people in Padova is working on generalizations of the work of Zanardo to classify length functions of Prufer domains.

The classification given by Vamos on Noetherian rings was generalized by him in his (non-pubblished) PhD thesis to a classification for rings with Gabriel-Krull dimension. I recently gave an alternative poof of Vamos' result for Grothendieck categories with Gabriel-Krull dimension based on the formalism of torsion theories, contact me if you are interested.

Let me suggest you some references:

• Northcott, D. G.; Reufel, M., A generalization of the concept of length, Q. J. Math., Oxf. II. Ser. 16, 297-321 (1965). Zbl 0129.02203.

• Vamos, Peter, Additive functions and duality over Noetherian rings, Q. J. Math., Oxf. II. Ser. 19, 43-55 (1968). Zbl 0153.37101.

• Zanardo, Paolo, Multiplicative invariants and length functions over valuation domains, J. Commut. Algebra 3, No. 4, 561-587 (2011). Zbl 1250.13016.

If your interest goes in this direction I know that people in Padova is working on generalizations of the work of Zanardo to classify length functions of Prufer domains.

The classification given by Vamos on Noetherian rings was generalized by him in his (non-pubblished) PhD thesis to a classification for rings with Gabriel-Krull dimension. I recently gave an alternative poof of Vamos' result for Grothendieck categories with Gabriel-Krull dimension based on the formalism of torsion theories, contact me if you are interested.