Geodesic
Direct sums of injective semimodules and direct products of projective semimodules over semirings
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2010), pp. 31-43

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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.
Keywords: semiring, injective semimodule, projective semimodule. 
@article{IVM_2010_10_a2,
     author = {S. N. Ilyin},
     title = {Direct sums of injective semimodules and direct products of projective semimodules over semirings},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {31--43},
     publisher = {mathdoc},
     number = {10},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_10_a2/}
}
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S. N. Ilyin. Direct sums of injective semimodules and direct products of projective semimodules over semirings. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2010), pp. 31-43. http://geodesic.mathdoc.fr/item/IVM_2010_10_a2/