Pseudo-projective Semi-modules And Quasi-injective Semi-modules

In this paper， the properties and applications of pseudo-projectivesemi-module， quasi-injective semi-module and max-projective semi-module，which are the important generalization of projective and injective semi-modules arefurther discussed．This paper is divided into four chapters：The first chapter is the preliminaries，We introduce some correspondingdefinitions and conclusions which are be used frequently．In the second chapter，we extend the definition of the pseudo-projective moduleto semi-modules， and probe its some properties． We also define thepseudo-projective covers of semi-module with universal properties．We also give theconcepts of the Pseudo-projective decomposition and the Pseudo-projectivedimension of semi-modules．On the basis of the third chapter, Chapter III introduces the properties of quasi-injective to semi-models, and also introduces the quasi-injective cover tosemi-module．Chapter III probes their properties．In the fourth chapter, we introduce the properties of pseudo-injective andM-pseudo–injective to semi-modules.In the fifth chapter, we introduce the conception of max-projective semi-module.And on the basis, the relationship of max-projective semi-module andHomR (M,)is described. Then it is proved that the direct sum of max-projectivesemi-modules is still max-projective semi-module.