Some Results Of The Semi-mode Theory

It is a common technique to study semiring through the semimodule. Many properties of semimodule are related to the faithful property of semi-module.This thesis consists of three chapters, which mainly study the annihilator of semimodule and the faithful property of semimodule.In chapter 1, we introduce the concepts of the R- balanced mapping, the R- bilinear mapping, and the tensor products defined upon these two types of mappings. We prove that the balanced tensor product is isomorphic to the bilinear tensor product when R is a commutatively associative ring with identity.The chapter 2 and 3 are the key part of this paper. After an introduction to the concepts of annihilator and annihilator on tensor product, we obtain a series of properties, such as how the annihilator changes under the monomor-phisms, epimorphisms and isomorphisms of semimodule. Following the an-nihilator and it's properties, we discuss the faithful and completely faithful semimodule and obtain some properties such as the direct sums and products preserve the properties of faith and complete faith of semimodule. Some other ways to describe the properties of faith are given. We prove that the free semi- modules, the generators and cogenerators in the category of semimodule are faithful.