a b s t r a c tThis paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. We provide some preliminaries on quasiprojective modules over commutative rings. Then we investigate the correlation with the well-known Prüfer conditions; that is, 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 Prüfer conditions between a ring and its total ring of quotients. We then examine various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasiprojectivity, marking their distinction from related classes of Prüfer rings.
The so called dense pairings were studied mainly by D. Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with J. Gómez-Torrecillas and J. Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A → C * . Such pairings, satisfying the so called α-condition, were called in the author's dissertation measuring α-pairings and can be considered as the corner stone in his study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g. Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Y. Doi , M. Koppinen and C. Menini et al.
Let A be an algebra over a commutative ring R. If R is noetherian and A• is pure in R A , then the categories of rational left A-modules and right A • -comodules are isomorphic. In the Hopf algebra case, we can also strengthen the BlattnerMontgomery duality theorem. Finally, we give sufficient conditions to get the purity of A• in R A .
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