An isomorphism test for modules over a non-commutative PID. Applications to similarity of Ore polynomials. (July 2016)

• Record Type: Journal Article
• Authors: Gómez-Torrecillas, José; Lobillo, F.J.; Navarro, Gabriel
• Is Part Of: Journal of symbolic computation. Volume 75(2016)
• Page Start: 149
• Page End: 170
• Publication Date: 2016-07
• Subjects: 16S36 -- 16Z05 -- 68W30; Isomorphic modules -- Similar Ore polynomials -- Canonical matrix form; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.11.012
• Languages: English
• ISSNs: 0747-7171

Abstract

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 / R f and R / R g . 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.