Strong Purity and Phantom Morphisms
Commutative Algebra
2024-10-15 v1
Abstract
Let be a commutative ring and be a multiplicative subset. We introduce and study the concept of -purity based on the notion of -strongly flat modules. The class of -pure injective modules will be studied. We demonstrate that this class is enveloping and explore its closedness under extension. The concept of purity is closely connected to the existence of phantom maps. So we will delve into the study of the -phantom morphisms. We will establish that the -phantom ideal is a precovering ideal and examine the situations where it becomes a covering ideal. Finally, in the last section, we will investigate an ideal version of the `Optimistic Conjecture', raised by Positselski and Sl\'{a}vik.
Cite
@article{arxiv.2410.09852,
title = {Strong Purity and Phantom Morphisms},
author = {R. Hafezi and J. Asadollahi and S. Sadeghi and Y. Zhang},
journal= {arXiv preprint arXiv:2410.09852},
year = {2024}
}