An isomorphism test for modules over a non-commutative PID. Applications to similarity of Ore polynomials

Abstract

Let R be a non-commutative PID finitely generated as a module over its center C. In this paper we give a criterion to decide effectively whether two given elements f,g∈R 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 also consider the more general problem of deciding when two given left R-modules of finite length are isomorphic. This criterion allows the design of algorithms when R is an Ore extension of a skew-field whose center is a commutative polynomial ring. We propose two methods which, essentially, check the equality of the rational canonical forms of certain matrices with coefficients in C associated to each of the modules. These algorithms are based on the fact that, if R is finitely generated as a C-module, then the existence of an isomorphism of R-modules can be reduced to checking the existence of an isomorphism of C-modules. Actually, we prove this result in the realm of non-commutative principal ideal domains, generalizing a version given by Jacobson for some Ore extensions of a skew field by an automorphism.