Characterizing Rings in Terms of the Extent of the Injectivity of Their Simple Modules

This work aims to contribute to the study of rings in terms of their simple modules. Vast research has been done in this direction to investigate rings whose simple modules are all injective, called V-rings. In this paper, we consider rings over which every simple module is either injective or furthest from being injective, in some certain senses. In this line of thinking, we pursue our investigation of rings under three main headlines, namely, rings with simple modules which are either injective or poor, rings with simple modules which are either injective or indigent, and rings with simple modules which are either injective or test modules for injectivity by subinjectivity. V-rings are an example of rings of all three types. Besides some interesting nontrivial examples, we provide several characterizations for rings of above types.