Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prufer conditions between a ring and its total ring of quotients. Section 4 examines various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prufer rings.Comment: 14 page

On topological lattices and their applications to module theory

Yassemi's "second submodules" are dualized and properties of its spectrum are studied. This is done by moving the ring theoretical setting to a lattice theoretical one and by introducing the notion of a (strongly) topological lattice L = (L, Lambda, V) with respect to a proper subset X of L. We investigate and characterize (strongly) topological lattices in general in order to apply it to modules over associative unital rings. Given a non-zero left R-module M, we introduce and investigate the spectrum Spec(f) (M) of first submodules of M as a dual notion of Yassemi's second submodules. We topologize Spec(f) (M) and investigate the algebraic properties of M by passing to the topological properties of the associated space

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

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

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