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 $g: B\rightarrow N$ and homomorphism $f : M \rightarrow N$, there exists a homomorphism $h: M\rightarrow B$ 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 FP-injective 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.