Injective projective and continuous modules and their relationships to some rings: A critical review

Abstract

Projective and Injective modules arise quite abundantly in nature. For example, all free modules that we know of, are projective modules. Similarly, the group of all rational numbers and any vector space over any field are examples of injective modules. In this paper, we study the theory of projective and injective modules. This expository note delves into the theory of projective modules parallel to the one developed for injective modules by Matlis. Given a perfect ring R, we present a characterization of indecomposable projective R-modules and describe a one-to-one correspondence between the projective indecomposable R-modules and the simple R-modules.