Linear systems over localizations of rings

TL;DR

This paper presents a method for solving linear systems over localizations of coherent rings, leveraging algorithms for ideal membership and syzygy computations, expanding computational algebra capabilities.

Contribution

It introduces a novel approach that combines ideal membership decision algorithms with syzygy computations to solve linear systems over localized rings.

Findings

Method works under coherence and ideal membership decision assumptions

Algorithm computes solutions over localizations effectively

Extends computational techniques to broader classes of rings

Abstract

We describe a method for solving linear systems over the localization of a commutative ring $R$ at a multiplicatively closed subset $S$ that works under the following hypotheses: the ring $R$ is coherent, i.e., we can compute finite generating sets of row syzygies of matrices over $R$, and there is an algorithm that decides for any given finitely generated ideal $I \subseteq R$ the existence of an element $r$ in $S \cap I$ and in the affirmative case computes $r$ as a concrete linear combination of the generators of $I$.