Applications of $\bigcap$-large pseudo N-injective acts in quasi-Frobenius monoid theory and its relationship with some classes of injectivity

The aim of this paper is to review thoroughly the applications of ∩ -large pseudo N-injective acts in quasi-Frobenius monoid theory, and therefore, the relationship of ∩ -large pseudo N-injective acts with some class of injectivity is studied. Applications of the properties of ∩ -large pseudo injective acts in quasi-Frobenius monoid theory are proven. Also, it’s proved that the subsequent parity, every union (direct sum) of the two ∩ -large pseudo injective acts is a ∩ -large pseudo injective act if and only if every ∩ -large pseudo injective act is injective under Noetherian condition for a right monoid S. Additionally, we proved that the category of strongly ∩ -large pseudo N-injective right S-acts are going to be egalitarian to the category of projective right S-acts under monoid conditions. The connections between quasi injective and ∩ -large pseudo injective acts are investigated.