# structure theorem for modules

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 .

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I'm guessing you're referring to the structure of modules over a principal ideal domain? A complete classification of modules over a polynomial ring in $2$ variables is already very difficult, but a lot is known. What sort of answer were you hoping for? – Donu Arapura Jun 14 '11 at 15:14
To add to Donu's comment in the spirit of unknown's answer: the finite-length modules over $k[x,y]$ are of wild representation type -- they can't be parametrized by the points of any finite-dimensional variety. Put another way, a classification of those modules would entail a classification of \emph{all} modules over \emph{every} finite-dimensional $k$-algebra. So: hopeless. The result about $k[x,y]$ is due to Drozd, I think. – Graham Leuschke Jun 14 '11 at 16:08
A reference for the result on $k[x,y]$ is [I. M. Gelfand and V. A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Functional Anal. Appl. 3 (1969) 325-326.] They prove that classifying artinian modules over $k[x,y]$ is as hard as classifying modules over a free algebra on three generators. Drozd's work implies this last problem is wild. – Mariano Suárez-Alvarez Jun 14 '11 at 21:08

## 1 Answer

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 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.

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