Rings with modules having a restricted injectivity domain

Abstract

We introduce modules whose injectivity domains are contained in the class of modules with zero radical and call them working-class. This notion gives a generalization of poor modules that have minimal injectivity domain. Semisimple working-class modules always exist for arbitrary rings whereas their predecessors do not. We investigate the rings over which every module is either injective or working-class. Right weakly V-rings are examples of these rings. Moreover, we study the existence of workingclass simple modules and show that if there is a projective working-class simple right module, then the ring is a right GV-ring.

On the Splitting Property of Rings with Restricted Class of Injectivity Domains

“…In Section 3, we focus on indecomposable rings and explore rings with local profiles. We also apply our findings to particular types of rings discussed in [2], [3], and [7]. We complete our work with a characterization of indecomposable rings with local profiles in terms of their quasi-injective modules (Theorem 3.8).…”

“…Note that the class V consists of modules whose factors have zero radicals. Thus, rings discussed in [7,Sec. 3] are V-local rings.…”

“…Since the inception of rings with no middle class in 2010, there has been a growing research interest in examining rings by imposing restrictions on the injectivity domains of their modules. In [3,4,5,7,10,19,20], rings whose modules are of restricted domains of injectivity are studied. Each of these papers is interested in when modules are either injective or far from being injective to a specified degree.…”

“…In [4], rings with local profiles whose unique coatom consists of all locally Noetherian V-modules are discussed. Also, in [7], rings with local profiles whose unique coatom is the class of all V-modules are considered. In each of these papers, the previously mentioned splitting property of the rings of interest has been demonstrated.…”

A Hyperstructural Approach to Semisimplicity

“…Moreover, in recent years, types of injectivity and projectivity have been studied with the help of (semi)simple modules. Refer to the books [10,11,[23][24][25] and the papers [26][27][28][29][30][31][32][33][34] for detailed information.…”