This article is cited in 17 papers

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

S. N. Ilyin

Chair of Algebra and Mathematical Logics, Kazan State University, Kazan, Russia

Abstract: 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.

UDC: 512.558

Received: 30.12.2008

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2010, 54:10, 27–37

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