A Constructive Approach to Arithmetics in Ore Localizations
Johannes Hoffmann, Viktor Levandovskyy · 2017
Classical results of Ore from the 1930's show that a localization of a domain R with respect to a multiplicatively closed set S exists if S satisfies the left Ore property. We investigate the arithmetics of the localized ring S-1R from both theoretical and practical points of view. A major obstacle is the computation of the intersection of a left ideal with a submonoid S of R, which can be overcome by requiring S to be equipped with additional structure. Therefore we distill three most frequently occurring types of Ore sets and provide partial solutions to the mentioned problem for these types. We provide an implementation of arithmetics over the ubiquitous G-algebras in Singular:Plural and discuss algorithmic and theoretic questions in this context.