On isomorphisms of modules over non-commutative PID

Abstract

Let R be an Ore extension of a skew-field. A basic computational problem is to decide effectively whether two given Ore polynomials f, g ∈ R (of the same degree) are similar, that is, if there exists an isomorphism of left R--modules between R/Rf and R/Rg. Since these modules are of finite length, we consider the more general problem of deciding when two given left R--modules of finite length are isomorphic. We show that if R is free of finite rank as a module over its center C, then this problem can be reduced to check the existence of an isomorphism of C--modules. This method works for a large class of left R--modules of finite length. Our result is proven in the realm of non-commutative principal ideal domains, and generalizes a result by Jacobson for some Ore extensions of a skew field by an automorphism. As a consequence, we propose an algorithm to check whether two given left R--modules of finite length are isomorphic by associating a matrix with coefficients in C to each of the modules, and checking if the corresponding rational canonical forms are equal. Our method is illustrated with examples of computations for Ore extensions of finite fields, and of the Hamilton quaternions.