Existence of covers and envelopes of a left orthogonal class and its right orthogonal class of modules
Umamaheswaran Arunachalam, Udhayakumar Ramalingam, Selvaraj Chelliah

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 -projective, -injective and -flat modules and give some characterizations of these modules, where is a class of left -modules. We prove that the class of all -projective modules is Kaplansky. Further, if the class of all -projective -modules is closed under direct limits, we show the existence of -projective covers and -injective envelopes over a -hereditary ring Moreover, we decompose a -projective module into a projective and a coreduced -projective module over a self -injective and -hereditary ring. Finally, we prove that every module has a -injective precover over a coherent ring where is the…
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Taxonomy
TopicsRings, Modules, and Algebras · Algebraic structures and combinatorial models · Advanced Topics in Algebra
