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A criterion for rings which are locally valuation rings

Kamran Divaani-Aazar, Mohammad Ali Esmkhani, Massoud Tousi (2009)

Colloquium Mathematicae

Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain R is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semisimple if and only if every R-module is cyclically pure...

A monogenic Hasse-Arf theorem

James Borger (2004)

Journal de Théorie des Nombres de Bordeaux

I extend the Hasse–Arf theorem from residually separable extensions of complete discrete valuation rings to monogenic extensions.

Component groups of abelian varieties and Grothendieck's duality conjecture

Siegfried Bosch (1997)

Annales de l'institut Fourier

We investigate Grothendieck’s pairing on component groups of abelian varieties from the viewpoint of rigid uniformization theory. Under the assumption that the pairing is perfect, we show that the filtrations, as introduced by Lorenzini and in a more general way by Bosch and Xarles, are dual to each other. Furthermore, the methods yield some progress on the perfectness of the pairing itself, in particular, for abelian varieties with potentially multiplicative reduction.

Formal prime ideals of infinite value and their algebraic resolution

Steven Dale Cutkosky, Samar ElHitti (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Suppose that R is a local domain essentially of finite type over a field of characteristic 0 , and ν a valuation of the quotient field of R which dominates R . The rank of such a valuation often increases upon extending the valuation to a valuation dominating R ^ , the completion of R . When the rank of ν is 1 , Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than 1 , there is no natural ideal in R ^ that...

Gaussian and Prüfer conditions in bi-amalgamated algebras

Najib Mahdou, Moutu Abdou Salam Moutui (2020)

Czechoslovak Mathematical Journal

Let f : A → B and g : A → C be two ring homomorphisms and let J and J ' be ideals of B and C , respectively, such that f - 1 ( J ) = g - 1 ( J ' ) . In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of A with ( B , C ) along ( J , J ' ) with respect to ( f , g ) (denoted by A ⋈ f , g ( J , J ' ) ) , introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties.

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