Existence of covers and envelopes of a left orthogonal class and its right orthogonal class of modules

TL;DR

This paper explores the existence and properties of covers and envelopes related to left orthogonal classes of modules, providing characterizations, decompositions, and conditions for their existence over specific rings.

Contribution

It establishes the existence of covers and envelopes for orthogonal classes of modules and characterizes their structure over various classes of rings.

Findings

The class of all $ ext{X}^ot$-projective modules is Kaplansky.

Existence of $ ext{X}^ot$-projective covers and $ ext{X}$-injective envelopes under certain conditions.

Decomposition of modules into projective and coreduced parts over specific rings.

Abstract

In this paper, we investigate the notions of $\mathcal{X}^\bot$-projective, $\mathcal{X}$-injective and $\mathcal{X}$-flat modules and give some characterizations of these modules, where $\mathcal{X}$ is a class of left $R$-modules. We prove that the class of all $\mathcal{X}^\bot$-projective modules is Kaplansky. Further, if the class of all $\mathcal{X}$-projective $R$-modules is closed under direct limits, we show the existence of $\mathcal{X}^\bot$-projective covers and $\mathcal{X}$-injective envelopes over a $\mathcal{X}^\bot$-hereditary ring $R.$ Moreover, we decompose a $\mathcal{X}^\bot$-projective module into a projective and a coreduced $\mathcal{X}^\bot$-projective module over a self $\mathcal{X}$-injective and $\mathcal{X}^\bot$-hereditary ring. Finally, we prove that every module has a $\mathcal{W}$-injective precover over a coherent ring $R,$ where $\mathcal{W}$ is the class of all pure projective modules.