An alternative perspective on pure-projectivity of modules

Abstract

The study of pure-projectivity is accessed from an alternative point of view. Given modules M and N, M is said to be N-pure-subprojective if for every pure epimorphism g ∶ B → N and homomorphism f ∶ M → N , there exists a homomorphism h ∶ M → B such that gh = f. For a module M, the pure-subprojectivity domain of M is defined to be the collection of all modules N such that M is N-pure-subprojective. We obtain characterizations for various types of rings and modules, including FPinjective and FP-projective modules, von Neumann regular rings and pure-semisimple rings in terms of pure-subprojectivity domains. As pure-subprojectivity domains clearly include all pure-projective modules, a reasonable opposite to pure-projectivity in this context is obtained by considering modules whose pure-subprojectivity domain consists of only pure-projective. We refer to these modules as psp-poor. Properties of pure-subprojectivity domains and of psp-poor modules are studied.

Rings Whose Pure-Projective Modules Have Maximal or Minimal Projectivity Domain

“…This provides the last case of Theorem 4. 10. But what about the case; where Λ is a semiprime ring and Soc(Λ Λ ) ̸ = 0.…”

“…Let R be an associative ring with identity throughout the article, and unless otherwise indicated, any module be a right R-module. Projectivity has been investigated from various angles in the recent studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The class {Y ∈ Mod-R : X is Y -projective} for a module X is referred to as the projectivity domain of X and is represented by Pr −1 (X) [16].…”