Uniformly S-projective relative to a module and its dual
Mohammad Adarbeh, Mohammad Saleh
TL;DR
The paper develops a framework of uniformly $S$-relative homological notions for modules over a commutative ring: $u$-$S$-projective and $u$-$S$-injective relative to a module, along with $u$-$S$-semisimple and $u$-$S$-quasi-projective/injective variants. It characterizes these notions via induced Hom maps and $u$-$S$-exactness, establishing equivalences that tie $M$ being $u$-$S$-semisimple to all modules being $u$-$S$-injective or $u$-$S$-projective relative to $M$, and similarly dual conditions for endomorphisms. The paper also provides a local characterization of quasi-projective/injective modules and demonstrates that $u$-$S$-quasi-projective/injective notions sit strictly between the classical and uniformly strong notions, with additional results on direct sums and the semisimplicity of rings. Overall, it offers a cohesive set of criteria linking relative homological properties to global ring/module structure under a multiplicative set $S$.
Abstract
In this article, we introduce the notion of u-S-projective relative to a module. Let S be a multiplicative subset of a ring R and M an R-module. An R-module P is said to be u-S-projective relative to M if for any u-S-epimorphism f : M \to N, the induced map HomR(P, f) : HomR(P,M) \to HomR(P,N) is a u-S-epimorphism. Dually, we also introduce u-S-injective relative to a module. Some properties of these notions are discussed. Several characterizations of u-S-semisimple modules in terms of these notions are given. The notions of u-S-quasi-projective and u-S-quasi-injective modules are also introduced and some of their properties are discussed.