An alternative perspective on projectivity of modules

TL;DR

The paper introduces the concept of relative subprojectivity to measure module projectivity differently, defining subprojectivity domains and exploring properties and existence of modules with minimal domains, especially over artinian serial rings.

Contribution

It proposes a new notion of relative subprojectivity and studies the properties and existence of modules with minimal subprojectivity domains.

Findings

Subprojectivity domains can be characterized for modules.

Existence of sp-poor modules is established for artinian serial rings.

Subprojectivity provides an alternative perspective on module projectivity.

Abstract

Similar to the idea of relative projectivity, we introduce the notion of relative subprojectivity, which is an alternative way to measure the projectivity of a module. Given modules $M$ and $N$, $M$ is said to be {\em $N$-subprojective} if for every epimorphism $g:B \rightarrow N$ and homomorphism $f:M \rightarrow N$, then there exists a homomorphism $h:M \rightarrow B$ such that $gh=f$. For a module $M$, the {\em subprojectivity domain of $M$} is defined to be the collection of all modules $N$ such that $M$ is $N$-subprojective. A module is projective if and only if its subprojectivity domain consists of all modules. Opposite to this idea, a module $M$ is said to be {\em subprojectively poor}, or {\em $sp$-poor} if its subprojectivity domain is as small as conceivably possible, that is, consisting of exactly the projective modules. Properties of subprojectivity domains and $sp$-poor modules are studied. In particular, the existence of an $sp$-poor module is attained for artinian serial rings.