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A class of torsion-free abelian groups characterized by the ranks of their socles

Ulrich F. Albrecht, Anthony Giovannitti, H. Pat Goeters (2002)

Czechoslovak Mathematical Journal

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket R -module is R tensor a bracket group.

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...

Birings and plethories of integer-valued polynomials

Jesse Elliott (2010)

Actes des rencontres du CIRM

Let A and B be commutative rings with identity. An A - B -biring is an A -algebra S together with a lift of the functor Hom A ( S , - ) from A -algebras to sets to a functor from A -algebras to B -algebras. An A -plethory is a monoid object in the monoidal category, equipped with the composition product, of A - A -birings. The polynomial ring A [ X ] is an initial object in the category of such structures. The D -algebra Int ( D ) has such a structure if D = A is a domain such that the natural D -algebra homomorphism θ n : D i = 1 n Int ( D ) Int ( D n ) is an isomorphism for...

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