On Closed Quasi Principally Injective Acts over Monoids

The concept of closed quasi principally injective acts over monoids is introduced ,which signifies a generalization for the quasi principally injective as well as for the closed quasi injective acts. Characterization of this concept is intended to show the behavior of a closed quasi principally injective property. At the same time, some properties of closed quasi principally injective acts are examined in terms of their endomorphism monoid. Also, the characterization of a closed self-principally injective monoid is given in terms of its annihilator. The relationship between the following concepts is also studied; closed quasi principally injective acts over monoids, Hopfian, co Hopfian, and directly finite property. Ultimately, based on the results obtained, the conditions on subacts to inherit a closed quasi principally injective property were shown. Part of this paper was dedicated to studying the relationship between the classes of closed quasi principally injective acts with some generalizations of injectivity. Conclusions and future remarks of this work are given.