Can structure theorem for modules be extended to modules over UFDS , to modules over Neotherian rings ? if yes then can one get the statement and reference?


Since operations on matrices with coefficients as polynomials in several variables some extension seems possible .



structure theorem for modules
Let me start by something classical: extending the classical result for PIDs, by Steinitz's Theorem (1912) all finitely generated modules over Dedekind domains are characterized, see <http://en.wikipedia.org/wiki/Dedekind_domains> and scroll down.


Beyond Dedekind domains things get complicated, but there is considerable recent work going on. The following [paper](http://www.math.wisc.edu/~levy/preprints/Levy%2DLisbon%2Dfinal.pdf) by Levy and Klingler gives an overview on their expansive (recent) work on this. Very roughly, there is a tame/wild dichotomy (very informally 'wild' is more or less 'hopeless') and they give an (essentially) complete answer for those noetherian rings were the answer is not 'wild'; the key-word here is Dedekind-like.
The 'essential' refers to the fact that, as they point out, there are or at least might be some exceptions in some characteristic two cases. 


I am convinced that I once saw some subsequent work on this particular cases (though I do not remember whether it was partial or complete), yet unfortunately I am unable to find it right now.


