Constructive Arithmetics in Ore Localizations with Enough Commutativity

Abstract

We continue the investigations of the constructivity of arithmetics within non-commutative Ore localizations, initiated in our 2017 ISSAC paper, where we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization. We provide algorithms to compute such closures for certain non-commutative rings with respect to Ore sets with enough commutativity.