Some Studies Of The Short Proper Exact Sequences Of Semimodules And Pure Injective Semimodules

In the paper, we mainly investigate the short proper exact sequence of semi-module and the pure injective semi-modules and the pure quasi-injective semi-module, than we got some properties. The paper falls into four parts.The first section is the preliminaries, we introduce some corresponding definitions and conclusions which are be used frequently.The second section, by some conclusion of exact sequence in module, and some conclusion of exact sequence under the view of congruence of semi-modules, then we generalize some properties of short exact sequence in the category of modules to semi- module category.The last section, in order to study the properties of pure semi-modules, we know some conclusions and definitions of pure projective semi-modules, and on this basis,we give the definitions of pure injective semi-modules and pure quasi-injective semi-module, and then discuss the relationship between pure injective semi-module and Hom(M,-).