Some Studies Of Essential Injective Semimodules And P-flat Semimodules

In the paper, we mainly investigate essential injective semimodules and P-falt semimodules. Discuss the properties of essential injective semimodules, essentialinjective dimension, the results of P-falt semimodules and k-P-falt semimodules. Describe the relationship of k-P-falt semimodules and P-injectivesemimodules. The paper falls into four parts.The first section is the preliminaries，we introduce corresponding definitionsand conclusions which are be used frequently．In the second section, we introduce the concept of essential injectivesemimodules. Some properties of it are given. Moreover, the essential injectiveresolution and essential injective dimension of semimodules M on theadditively-idempotent semiring are defined.The characteristics of the semimodulesare discussed for essential injective dimension0and1.In the third section, the concept of prime injective semimodules is introducedby the theory of homological. We gain the equivalent characters of prime injectivesemimodules and study that the direct sum of prime injective semimodules are stillprime injective semimodules under the condition of every prime ideal of semiring Rwhich is finitely generated.In the fourth section, we define P-falt semimodules and k-P-falt semimodules, and discuss their related properties. Secondly describe the relationshipof k-P-falt semimodules and P-injective semimodules. In the end, the leftprincipal ideal of semiring R is studied by means of k-P-falt rightR-semimodules.