Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that, in the case of injectivity of direct sum or projectivity of direct product of a family of semimodules over a semiring $S$, a subfamily consisting of all semimodules of a family which are not modules is either finite or has a cardinality strictly lesser than a cardinality of a semiring $S$. As a consequence we obtain semiring analogs of known characterizations of classical semisimple, quasi-Frobenius, and one-side Noetherian rings.