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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Direct sums of injective semimodules and direct products of projective semimodules over semirings},
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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/

[1] Golan J. S., Semirings and their applications, Kluwer Academic Publishers, Dordrecht–Boston–London, 1999 | MR

[2] Kash F., Moduli i koltsa, Mir, M., 1981 | MR

[3] Feis K., Algebra: koltsa, moduli i kategorii, v. 2, Mir, M., 1979 | MR

[4] Skornyakov L. A., Elementy obschei algebry, Nauka, M., 1983 | MR | Zbl

[5] Ilin S. N., “Polukoltsa, nad kotorymi vse polumoduli in'ektivny (proektivny)”, Matem. vestn. pedvuzov i un-tov Volgo-Vyatsk. regiona, 2006, no. 8, 50–53