Several Properties Of Injective Semimodules

The semimodules on the semirings are the natural promotion of the modules on the ring. Although they have the similar structures of the module, but the differences are existed between them. In this paper, the special injective property is introdued to the semimodules on the semirings. We hope we can reserve the most of the good properties of the modules. Then In this paper, we generalize the Schanuel Lemma in the category of modules to injective semimodules, and we introduce the concepts of injective decomposition and injective dimension of semimodules and obtain some related results of them. The paper falls into three parts.The first section is preliminary .We introduce some related definitions and results for later parts .In the second section, the notion of P - injective semimodules is introduced. And on the basis, the relationship of P - injective semimodules and HomR (- ,E) is described. Then it is proved that the direct sum and the direct product of P - injective semimodules are still P - injective semimodules. In addition, the relationship of P - injective semimodules and divisible semimodules is decribed in division semirings. Finally as the conditions of P - injective semimodules' notion are weaker than the notion of injective semimodules, it is proved that non-zero P - injective semimodules are existed over arbitrary proper semirings.In the third section, the notions of GP - injective semimodules and n - P-injective semimodules are introduced. Not only the relationship of GP -injective semimodules and generalized PP - semirings is given, but also the relationship of n - P-injective semimodules and n- generalized PP - semirings is given.In the four section, the concepts of injective decomposition and injective dimension of semimodules are introduced. And the Schanuel Lemma in the category of the injective modules is promoted to the injective semimodules.