Uniformly S-pseudo-injective modules
Mohammad Adarbeh, Mohammad Saleh
TL;DR
The paper generalizes injectivity concepts via a uniform $S$-torsion scaling, introducing $u$-$S$-pseudo-injective modules and clarifying their position between $u$-$S$-injective and $u$-$S$-quasi-injective modules. It establishes fundamental properties, including stability in $u$-$S$-split exact sequences, direct-sum behaviour, and a key equivalence: $M$ is $u$-$S$-quasi-injective iff $M^{(2)}$ is $u$-$S$-pseudo-injective. It then defines two related ring classes, $Qu$-$S$-$I$-rings and $u$-$S$-$Qu$-$S$-$I$-rings, and proves a hierarchical chain with $QI$-rings and $u$-$S$-semisimple rings, along with local characterizations. Together, these results yield criteria for when every module is $u$-$S$-pseudo-injective or when such modules are forced to be $u$-$S$-injective, advancing classification of rings and module categories under $S$-torsion constraints.
Abstract
This paper introduces the notion of uniformly-S-pseudo-injective (u-S-pseudo-injective) modules as a generalization of u-S-injective modules. Let R be a ring and S a multiplicative subset of R. An R-module E is said to be u-S-pseudo-injective if for any submodule K of E, there is s in S such that for any u-S-monomorphism f : K \to E, sf can be extended to an endomorphism g : E \to E. Several properties of this notion are studied. For example, we show that an R-module M is u-S-quasi-injective if and only if M \oplus M is u-S-pseudo-injective. Two classes of rings related to the class of QI-rings are introduced and characterized.