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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.

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 Singular:Plural and discuss questions arising in this context. Numerous examples illustrate the effectiveness of the proposed approach.

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