Abstract
In act theory, Pseudo injective acts and their generalizations are essential. As a result, the purpose of this work is to give a generalization of pseudo quasi principally injective acts specifically small pseudo quasi principally injective acts. Besides, the concept of a small pseudo injective was presented, which can be employed later. If each S-monomorphism from a small M-cyclic sub-act of M\textsubscript{S} to N\textsubscript{S} is extended to S-homomorphism from M\textsubscript{S} to N\textsubscript{S}, An S-act N\textsubscript{S} is called a small pseudo-M-principally-injective (for simply small pseudo-MP-injective). Also, if an S-act M\textsubscript{S} is a small pseudo-MP-injective act, then it is called small pseudo quasi principally injective. This form of generalization is given several new characterizations and properties. Following that conditions are shown under which sub acts inherit the property of small pseudo quasi principally injectivity. Furthermore, the connection between the classes of small pseudo quasi principally injective S-acts with classes of injectivity is addressed, and as a result, requirements to coincide these classes are shown. Our work's conclusions have been explained.
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