Direct sums of injective semimodules and direct products of projective semimodules over semirings. (Q619496)

Let \(S\) be a semiring. The paper deals with semimodules over \(S\). The main results in the paper are: 1. If the direct sum of a family of semimodules over \(S\) is an injective semimodule or the direct product of a family of semimodules over \(S\) is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of \(S\). 2. As a consequence, the semiring analogs of the well-known characterizations of classical semisimple, quasi-Frobenius and one-sided Noetherian rings are obtained.