# Left saturation closure for Ore localizations

## TL;DR

This paper introduces the concept of left saturation closure (LSat) for modules at subsets of rings, unifying various notions related to Ore localization and aiding in understanding units and normal forms in localized rings.

## Contribution

It defines LSat, a new generalization that connects Ore localization, algebraic analysis, and module theory, providing a unified framework for saturation and localization.

## Key findings

- LSat characterizes units in localized rings
- Provides a normal form for left Ore sets
- Unifies concepts in Ore localization and algebraic analysis

## Abstract

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.