  Injective modules play an important role in characterizing different classes
of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no
non-zero injective semimodules (e.g. the semiring of non-negative integers). In
this paper, we study some of the basic properties of the so called e-injective
semimodules introduced by the first author using a new notion of exact
sequences of semimodules. We clarify the relationships between the injective
semimodules, the e-injective semimodule, and the i-injective semimodules
through several implications, examples and counter examples. Moreover, we
provide partial results for the so called Embedding Problem (of semimodules in
injective semimodules).
