On the Weak-Injectivity Profile of a Ring


AYDOĞDU P. , Lopez-Permouth S. R. , Sandoval-Miranda M. L. S.

BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume:
  • Publication Date: 2020
  • Doi Number: 10.1007/s40840-020-00938-3
  • Title of Journal : BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY

Abstract

We consider two types of weak-injectivity domains of a module. Small weak-injectivity domains gauge the extent of weak injectivity of modules; weak-injectivity domains are an alternative way to gauge their injectivity. We focus on notions which are opposite, respectively, to weakly injective or injective modules according to these two schemes. While we do consider both notions to some extent, our emphasis is on modules with smallest weak-injectivity domain; we name those modules extremely poor. The intersection of all (small) domains of weak injectivity is the natural focus of attention. The intersection of all (small) domains of weak injectivity contains the class of all (finitely generated) semisimple modules, but it is not always equal to it. The domain of weak injectivity of an extremely poor module thusly depends on the ring in question. We consider necessary and sufficient conditions for said intersections to consist solely of semisimple modules and look at examples of both circumstances, when this is the case and when it is not. When the intersection of weak-injectivity domains consists of semisimple modules, every extremely poor module is poor; we study further connections between these two notions. Comparisons are also drawn to paupers (poor modules with no proper poor direct summands); paupers are a type of poor module that is significant in understanding the intrinsic structure of poor modules. Extremely poor modules over PCI domains and extremely poor abelian groups are given particular attention.