Department of Mathematics Server University of Manitoba

Abstract: A well known result, a standard part of the structure theory of modules over a principal ideal domain, is the following:

Let A be matrix over a principal ideal domain D . Then there are invertible matrices X and Y over D such that A'=XAY is diagonal, and A' is essentially unique.

As a consequence, the formulas of the first order language of D-modules can be put into a particularly nice standard form, and as a result the model theory of D-modules is well understood.

I will give the (easy) proofs of these results.

Problem: Let A be the first Weyl algebra over a field of characteristic 0 . Then every left or right ideal is generated by no more than two elements. Is there any sort of standard form theorem for matrices over A ? Is there any sort of useful standard form theorem for the first order language of A-modules? (It is already known that the theory of A-modules is ``wild''.)