A Constructive Approach to Arithmetics in Ore Localizations

Hoffmann, Johannes; Levandovskyy, Viktor

doi:10.1145/3087604.3087649

Cited by 5 publications

(4 citation statements)

References 13 publications

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“…An implementation of basic arithmetic in Ore-localized G-algebras is available for certain special cases of the types defined in Definition 11.5 in the library olga.lib for the computer algebra system Singular:Plural ([6]), including the localizations in Example 11.6. Details can be found in [9].…”

Section: Discussionmentioning

confidence: 99%

Left saturation closure for Ore localizations

Hoffmann,

Levandovskyy

2019

Preprint

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In this paper, we introduce the notion of LSat, the left saturation closure of a subset of a module at a subset of the base ring, which generalizes multiple important concepts related to Ore localization. We show its significance in finding a saturated normal form for left Ore sets as well as in characterizing the units of a localized ring.Furthermore, LSat encompasses the notion of local closure of submodules and ideals from the realm of algebraic analysis, where it describes the result of extending a submodule or ideal from a ring to its localization and contracting it back again.

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Section: Discussionmentioning

confidence: 99%

Left saturation closure for Ore localizations

Hoffmann,

Levandovskyy

2019

Preprint

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“…In our paper Hoffmann and Levandovskyy (2017a) we have shown that this problem is integral to a constructive treatment of the Ore condition in G-algebras which in turn allows us to perform basic arithmetic operations in Ore localizations of G-algebras.…”

Section: Algorithmic Toolboxmentioning

confidence: 99%

“…This paper is a part of our broad program dedicated to realizing this approach. Starting point was the investigation of arithmetic operations with left and right fractions in Ore localizations of non-commutative domains in Hoffmann and Levandovskyy (2017a) and its extended version Hoffmann and Levandovskyy (2017b). We have demonstrated that such arithmetic operations are based essentially on two algorithms, namely…”

Section: Introductionmentioning

confidence: 99%

Constructive arithmetics in Ore localizations enjoying enough commutativity

Hoffmann

Levandovskyy

2021

Journal of Symbolic Computation

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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.

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“…This paper is an extended, expanded and enhanced version of the paper [10], which appeared at ISSAC 2017 in Kaiserslautern, Germany. Proofs have been either restored from the abridged version or expanded in details.…”

Section: Introductionmentioning

confidence: 99%

Constructive arithmetics in Ore localizations of domains

Hoffmann

Levandovskyy

2020

Journal of Symbolic Computation

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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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A Constructive Approach to Arithmetics in Ore Localizations

Cited by 5 publications

References 13 publications

Left saturation closure for Ore localizations

Left saturation closure for Ore localizations

Constructive arithmetics in Ore localizations enjoying enough commutativity

Constructive arithmetics in Ore localizations of domains

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