B. Torrecillas

Morita Duality for Grothendieck Categories with Applications to Coalgebras

ABSTRACT Finiteness conditions of reflexive objects of a Morita duality of Grothendieck category is studied. It is observed that the relation between coproduct and product is a fundamental fact. We show that for coalgebras with a duality, the class of reflexive objects coincides with the class of quasi-finite comodules.

On the existence of covers by injective modules relative to a torsion theory

Let R be a ring with identity. In this note we study covers of left R-modules by r-injectives left R-modules, where r is a hereditary torsion theory defined in the category of all left R-modules and all R-morphisms. When R is an artinian commutative ring, a complete answer about the existence of such covers for every R-module is given. In case that T is a centrally splitting torsion theory, we can characterize those T for which every left R-module has a T-injective cover. Also we analyze R-modules such that the injective and the T-injective cover are the same. At the end of this note we relate the concepts of colocalization and cover

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, a left H-comodule algebra 𝔅 and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules C ℳ(H)𝔅 in terms of modules. We will also show that for an H-bicomodule algebra 𝔸 and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules 𝔸𝒴𝒟(H) C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.

Graded Semiartinian Rings: Graded Perfect Rings

Abstract We study graded left semiartinian rings with finite support. It is shown that the semiartinian property is preserved when we pass to the smash product in the sense of Quinn. We apply these results to investigate left perfect graded rings.

On actions of Hopf algebras with cocommutative coradical

Let H be an m-dimensional Hopf algebra with left integral t, let R be a left H -module algebra with 1 containing an elementwith t � = 1, and let S =R H. Itis proved t R is fully integral over S, every simple right R-module has a length ≤ m over S and J (S) m ⊆J (R) ∩ S ⊆J (S), where J (R) is the Jacobson radical of R, provided that H is pointed. Finally, it is shown that if S is a PI algebra, then R is a PI algebra as well, provided that H has a cocommutative coradical. c

ON THE EXISTENCE OF FLAT COVERS IN R-gr

Recently, a proof of the existence of a flat cover of any module over an arbitrary associative ring with unit has been finally given (see 4-5). In this paper we prove the existence of flat covers in the category of graded modules over a graded ring. Some graded theoretical machinery is introduced to make the proof possible and new graded homological tools are developed.

Coflat monomorphisms of coalgebras

Abstract We consider left coflat monomorphisms of coalgebras, and establish a 1-1 correspondence between the set of isomorphism classes of left coflat monomorphisms, the set of some coidempotent subcoalgebras and the set of equivalence classes of perfect localization bicomodules as well.

Gabriel Dimension for Graded Rings

Using techniques of localization for Grothendieck categories with a family of projective generators, we show that for a graded ring R = ⊕σ∈GRσ with finite support if Re has Gabriel dimension then

Multiplication objects in commutative grothendieck categories

R\hbox{-}{\rm gr}

The picard groups of coalgebras

has Gabriel dimension. Moreover, adding some lattice results, we prove that if