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## 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 2011 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 2011 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 2011 at 21:08