Some Studies Of P-Projective Semimodules

In the paper we mainly investigate p-projective semimodules and p-flat semimodules. Discuss the properties of p-projective semimodules, p-projective dimension, p-projective cover, p-flat semimodules and k-p-flat semimodules.The paper falls into three parts.The first section is preliminary. We introducing some definitions and results which are be used for later parts.In the second section, firstly, we introduce the concept of p-projective semimodules. Then the relationship of p-projective semimodules E and functor HomR(E,-) are discussed. After that we get the equivalent characterizations about p-projective semi-modules. Then we extend the Schanuel Lemma from projective semi-modules to p-projective semi-modules. We define p-projective decomposition and p-projective dimension and discuss their related properties. The last, we define p-projective cover.In the third section, on the basis of exact sequence and tensor product, we introduce the concept of p-flat semimodules and k-p-flat semimodules and discuss their properties.