Constructive Arithmetics in Ore Localizations Enjoying Enough Commutativity

TL;DR

This paper advances constructive methods for arithmetic in non-commutative Ore localizations, especially when the Ore sets exhibit sufficient commutativity, and provides algorithms for ideal closure computations.

Contribution

It extends the classification of Ore localizations to rings with zero divisors and develops algorithms for arithmetic and ideal closure in these settings.

Findings

Arithmetic in localizations of commutative polynomial algebras is constructive.

Algorithms for computing symbolic powers of ideals are provided.

Methods for local closures in certain non-commutative rings are introduced.

Abstract

This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier 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, and present an algorithm for the computation of the symbolic power of a given ideal in a commutative ring. We also provide algorithms to compute local closures for certain non-commutative rings with respect to Ore sets with enough commutativity.