Constructive Arithmetics in Ore Localizations of Domains

For a non-commutative domain $R$ and a multiplicatively closed set $S$ the (left) Ore localization of $R$ at $S$ exists if and only if $S$ satisfies the (left) Ore property. Since the concept has been introduced by Ore back in the 1930's, Ore localizations have been widely used in theory and in applications. We investigate the arithmetics of the localized ring $S^{-1}R$ from both theoretical and practical points of view. We show that the key component of the arithmetics is the computation of the intersection of a left ideal with a submonoid $S$ of $R$. It is not known yet, whether there exists an algorithmic solution of this problem in general. Still, we provide such solutions for cases where $S$ is equipped with additional structure by distilling three most frequently occurring types of Ore sets. We introduce the notion of the (left) saturation closure and prove that it is a canonical form for (left) Ore sets in $R$. We provide an implementation of arithmetics over the ubiquitous $G$-algebras in \textsc{Singular:Plural} and discuss questions arising in this context. Numerous examples illustrate the effectiveness of the proposed approach.