"A Normal Form Algorithm for Matrices over k[x,y]/". Reinhard Laubenbacher rlaubenb@nmsu.edu Reinhard Laubenbacher New Mexico State University Department of Mathematical Sciences Las Cruces, NM 88003-0001 USA ABSTRACT: The classification of finitely generated modules over a principal ideal domain is one of the fundamental results of algebra. Every such module can be written uniquely, up to order of summands, as a direct sum of indecomposable modules which are either copies of the ring or are quotients of the ring modulo a power of a prime ideal. If the module is presented as the cokernel of a matrix then one can find this decomposition by computing the Smith normal form of the matrix. In this talk the Smith normal form algorithm is generalized to the ring k[x,y]/ of polynomials in x and y without mixed terms. As an application one obtains an algorithmic classification of finitely generated modules over k[x,y]/, and related rings such as certain types of group rings. Looking at the problem from a different point of view, one obtains an algorithmic classification of pairs of mutually annihilating linear operators on a finite dimensional vector space.