Rings whose mininjective modules are injective
Yusuf Alagöz, Sinem Benli-Göral, Engin Büyükaşık, Juan Ramón García Rozas, Luis Oyonarte
TL;DR
The paper characterizes rings $R$ for which every mininjective right $R$-module is injective, tying mininjectivity to min-flatness and simple-injectivity through the framework of right quasi $V$-rings, GV-rings, and Kasch rings. It proves that such rings are precisely right Noetherian rings with every min-flat left $R$-module projective, equivalently right Artinian, right strongly min-coherent, and right quasi $V$-ring, with further equivalences for Kasch and PS rings and a UT matrix case; it also furnishes an Ikeda-type characterization of $QF$-rings using only right mininjectivity. The results extend the landscape of ring and module theory by linking mininjectivity to classical classes and by providing concrete structural criteria and new indigent-module constructions. These contributions offer a unified, structurally grounded view of when mininjective modules collapse to injective modules and their implications for well-studied ring categories.
Abstract
The main goal of this paper is to characterize rings over which the mininjective modules are injective, so that the classes of mininjective modules and injective modules coincide. We show that these rings are precisely those Noetherian rings for which every min-flat module is projective and we study this characterization in the cases when the ring is Kasch, commutative and when it is quasi-Frobenius. We also treat the case of $n\times n$ upper triangular matrix rings, proving that their mininjective modules are injective if and only if $n=2$. We use the developed machinery to find a new type of examples of indigent modules (those whose subinjectivity domain contains only the injective modules), whose existence is known, so far, only in some rather restricted situations.