Proper-exact Sequence Of Semi-modules And It's Related Properties

Semi-ring is one of the most natural algebraic structure which people feel,because we add the usual addition and multiplication to the natural numbers can form a semi-ring.The Semi-module is a generalization of the mold,but this special algebra structure which also has semi-ring structure and the mode structure,has not been adequately studied.Currently,the semi-ring and the semi-modules on the semi-ring have been applied in mathematics and the oretical computer science as an important tool,and it is an effective way to use semi-modules to characterize semi-ring,so the research of semi-modules is natural and necessary.This paper describes the proper-exact sequences of semi-modules and its related properties,such as"the five lemma of semi-modules", special cases which can make the semi-modules to be splited in these special cases ,finitely per formant semi-modules .This paper is divided into four chapters:The first chapter is the preliminaries,We introduce some corresponding definitions and conclusions which are be used frequently.In the second chapter,we first give the concept of exact sequence and proper-exact sequence which belong to the semi-module category ,and on this basis,we prove some simple properties about the exact sequence mode and the proper-exact sequence on semi-modules.On the basis of the second chapter, Chapter III and then shows a semi- model of the "five lemma" and the "three Lemma", and discussed some special cases which can make the semi-modules to be splited in these special cases .In the fourth chapter, we first give the concept of finitely per formant semi-modules in half-module category, and then use the properties and theorems of second and third chapters which have been proven to derive the characterizations of finitely per formant semi-modules and the its important properties.