Generalization of injective S-acts

The injective topic is well known for its influential relevance in module theory and scholars have worked hard to identify generalizations for it. One of these generalizations is M-Mininjective and mininjective, so we extend these notions to S-act theory as well, since act theory represents the generalization of module theory. If every S-homomorphism from a simple M-cyclic subact of M-S into N-S can be extended to M-S, an S-act N-S is called M-mininjective. An S-act M-S is referred to as mininjective, if for each simple right ideal A of S and every S-homomorphism from A into M-S can be extended to S-homomorphism from S into M-S. We looked at the properties and characterizations of S-act where all subacts are M-cyclic and simple and all subacts are merely simple. These topics are shown using examples. With the provided concepts, we were able to accomplish improved results, obtaining novel characterizations of mininjective acts in terms of duality conditions. Additionally, the conditions under which subacts inherit the mininjective and M-mininjective properties are studied. The connection between the act of maximal right ideals of S and the act of minimal subact of T-M is explicated. Finally, the conditions under which the classes of M-mininjective acts and the classes of mininjective S-acts will coincide are defined. Our work's conclusions have been explained.