Correspondences of coclosed submodules

TL;DR

This paper explores a bijective correspondence between coclosed submodules in lattices connected by Galois relations, with applications to module theory and endomorphism rings.

Contribution

It establishes a new order-preserving bijection between coclosed submodules in related lattices and modules, extending previous module-theoretic results.

Findings

Bijection between coclosed elements in lattices via Galois connections

Correspondence between coclosed submodules of modules and their endomorphism rings

Application to finitely generated quasi-projective modules

Abstract

We establish an order-preserving bijective correspondence between the sets of coclosed elements of some bounded lattices related by suitable Galois connections. As an application, we deduce that if $M$ is a finitely generated quasi-projective left $R$-module with $S=End_R(M)$ and $N$ is an $M$-generated left $R$-module, then there exists an order-preserving bijective correspondence between the sets of coclosed left $R$-submodules of $N$ and coclosed left $S$-submodules of $Hom_R(M,N)$.