1 research outputs found
Constructive Arithmetics in Ore Localizations of Domains
For a non-commutative domain and a multiplicatively closed set the
(left) Ore localization of at exists if and only if 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 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 of . It is not known yet, whether there exists an algorithmic
solution of this problem in general. Still, we provide such solutions for cases
where 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
. We provide an implementation of arithmetics over the ubiquitous
-algebras in \textsc{Singular:Plural} and discuss questions arising in this
context. Numerous examples illustrate the effectiveness of the proposed
approach.Comment: 24 page