I-Injective Semimodules

In this paper we investigate particular injectiviation of semimodule and some results are proved which lead to the characterization of these i-injective semimodules.The paper falls into two parts.In the first section,the notion of i-injective semimodules is introduced and we prove that there exist non-zero i-injective semimodules over arbitrary proper semirings.In the second section,we give the notion of weakly injective semimodules and gain two equivalent characters:left R-semimodule E is i-injective semimodule if and only if Hom_R(-,E) keep the proper exact sequence;left R-semimodule E is injective semimodule if and only if E is weak injective semimodule and i-injective semimodule.At last,in the third section,we introduce the domain of i-injective semimodules and get the result that the domain of i-injective semidule is closed for substract subsemimodules and factor semimodules.