Some Studies Of Proper Projective Dimension And Pure Projective Over Semimodule

In this paper we mainly investigate pure projective semimodules and flat semimodules. Discuss the problems of proper projective resolution and proper projective dimension and give some relative results. The paper falls into four parts.The first section is preliminary. We introduce some related definitions and results for later parts.In the second section. The notion of proper projective resolution and proper projective dimension over semimodule are introduced in reference of the theory of projective modules and projective dimension . And we made a preliminary discussion to application of proper projective dimension on semiring and semimoudle.In the third section. In order to study property of pure projective on semiring, Firstly, pure subsemimodule and pure proper exact sequence are defined and give some relative results. Secondly, the notion of pure projective semimodule is introduced, and on the basis, discuss the relationship between the pure projective semimodule and functor Hom(P,-) .In addition, the characters of direct sum and split have been proved. Finally, investigate the relationship between the pure projective semimodule and pure projective module, and get the result that pure projective semimodule is pure projective module on the complete pure semiring.At last, In the four section, the notions of flat semimodules and K-flat semimodules are introduced, and study these notions structure and properties.In the end ,we discuss the relationship of K-flat semimodules and projective semimodules and injective semimodules.