84 research outputs found
On p-Schreier varieties of semimodules
In this article, we consider categories of all semimodules over semirings which are p-Schreier varieties, i.e., varieties whose projective algebras are all free. Among other results, we show that over a division semiring R all semimodules are projective iff R is a division ring, prove that categories of all semimodules over proper additively π-regular semirings are not p-Schreier varieties (in particular, this result solves Problem 1 of Katsov [8]), as well as prove that categories of all semimodules over cancellative division semirings are, in contrast, p-Schreier varieties. © Taylor & Francis Group, LLC
On Serre's Problem on Projective Semimodules over Polynomial Semirings
Among other results of this paper, we single out the following ones. We show that division rings are the only division semirings over which the categories of semimodules are Schreier varieties, i.e., all subsemimodules of free semimodules are free too. We a complete description of division semirings R over which the categories of semimodules ℳR are p-Schreier varieties, i.e., varieties whose all projective algebras are free. We give a complete description of proper division semirings R whose categories of semimodules ℳR(X) over the polynomial semirings R(X) over R, in not necessary commuting variables X, are p-Schreier varieties. We show that the categories of semimodules ℳR(X) over the polynomial semirings R(X) over N-valued semirings R, in particular ℳN(X), are p-Schreier varieties. We also show that for N-valued semirings S, the semimodule categories ℳS never are Schreier varieties. © 2014 Copyright Taylor & Francis Group, LLC
Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective
© 2016 Elsevier Inc.In this paper, we introduce homological structure theory of semirings and CP-semirings — semirings all of whose cyclic semimodules are projective. We completely describe semisimple, Gelfand, subtractive, and anti-bounded, CP-semirings. We give complete characterizations of congruence-simple subtractive CP-semirings and congruence-simple anti-bounded semirings, which solve two earlier open problems for these classes of semirings. We also study in detail the properties of semimodules over Boolean algebras whose endomorphism semirings are CP-semirings; and, as a consequence of this result, we give a complete description of ideal-simple CP-semirings
Semiring and semimodule issues in MV-algebras
In this paper we propose a semiring-theoretic approach to MV-algebras based
on the connection between such algebras and idempotent semirings - such an
approach naturally imposing the introduction and study of a suitable
corresponding class of semimodules, called MV-semimodules.
We present several results addressed toward a semiring theory for
MV-algebras. In particular we show a representation of MV-algebras as a
subsemiring of the endomorphism semiring of a semilattice, the construction of
the Grothendieck group of a semiring and its functorial nature, and the effect
of Mundici categorical equivalence between MV-algebras and lattice-ordered
Abelian groups with a distinguished strong order unit upon the relationship
between MV-semimodules and semimodules over idempotent semifields.Comment: This version contains some corrections to some results at the end of
Section
On V-semirings and semirings all of whose cyclic semimodules are injective
© Taylor & Francis Group, LLC. In this article, we introduce and study V-and CI-semirings—semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe Vsemirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and antibounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple antibounded CI-semirings which solve two earlier open problems for these classes of CI-semirings
Toward homological characterization of semirings by e-injective semimodules
© 2018 World Scientific Publishing CompanyIn this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective
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