Abstract
Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no nonzero 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 [Formula: see text]-injective semimodule, and the [Formula: see text]-injective semimodules through several implications, examples and counter examples. Moreover, we show that every semimodule over an arbitrary semiring can be embedded in a [Formula: see text]-[Formula: see text]-injective semimodule.
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