Some Studies Of Injective Semimodules And Cancellative Semimodules

Semiring is a algebraic construction between semigroup and ring. It issimilar with the construction of ring and module. We correspondingly establishprojective semimodules、injective semimodules、free semimodules、cancellativesemimodules, and got a lot of good properties.In this paper we mainly investigate the constructions and properties of thesemimodules on the semirings. In the first section, we give some pre-knowledge aboutsemirings and left-R semimodules, to prepare for the next chapter.Schanule Lemma as an important conclusion on the rings and modules,effectively helping us to describe the construction. In the second chapter, we continueto use the definition of proper to extend the Schanuel Lemma to injective semimodule,and extend it further more in the form of multi-injective semimodules bymathematical induction.In the third chapter, the concepts of completely subtractive injectivedecomposition and completely subtractive injective dimension of cancellativesemimodule are introduced in reference of the theory of injective modules andinjective dimension.The most important theorem in the fourth chapter is the equivalent description ofk-projective semimodule.