Some Properties of Strongly Principally Self-Injective Modules

The idea of generalizing quasi injective by employing a new term is introduced in this paper. The introduction of principally self-injective modules, which are principally self-injective modules. A number of characteristics and characterizations of such modules have been established. In addition, the idea of strongly mainly self-pure sub-modules was added, which is similar to strongly primarily self-injective sub-modules. Some characteristics of injective, quasi-injective, principally self-injective, principally injective, absolutely self-pure, absolutely pure, and finitely R -injective modules being lengthened to strongly principally self-injective modules. So, in the present work, some properties are added to the concept in a manner similar to the absolutely self-neatness. The fundamental features of these concepts and their interrelationships are linked to the conceptions of some rings. (Von Neumann) regular, left SF-ring, and left pp-ring rings are described via such concept. For instance, the homomorphic picture of every principally injective module be strongly principally self-injective if R being left pp-ring, and another example for a commutative ring R of every strongly principally self -injective module be flat if R being (Von Neumann) regular. Also, a ring R be (Von Neumann) regular if and only if each R -module being strongly principally