On duality theory and AB5 ∗ modules

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.

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

In this paper, we introduce and study the properties of chain conditions on M-cyclic submodules defined as M-Noetherian and M-Artinian modules and rings. We prove that the property of being iso-M-Noetherian (iso-M-Artinian) is inherited by submodules, quotient modules, and finite sums as well as finite direct sums. Finally, we prove the Hilbert basis theorem for iso-M-Noetherian (iso-M-Artinian) and verify that the property of modules of being M-Noetherian (M-Artinian) and iso-M-Noetherian (iso-M-Artinian) are Morita invariant.

As a generalization of QF rings, the notion of rings was introduced. Many authors have studied connections between rings, Morita duality and quotient rings. Basic part of these works can be found in [21]. Several results for rings are extended to more general classes of rings. Rings R whose injective hull E(RR) is torsionless (right QF-3' rings) and rings R in which every finitely generated submodule of the injective hull E(RR) is torsionless (according to (91, we call such rings right QF-3 rings) are such kind of rings. These rings are closely related to the dual functors, the Lambek torsion theories, and Morita duality for Grothendieck categories (see [5], [6], [7], [9] and [ID]). On the other hand, these rings are generalized to modules, as QF-3' and QF-3 modules. QF-3' modules were mainly investigated from torsion theoretical aspects in [2] and [13]. Recently QF3 modules were introduced and studied by Ohtake in [16]. He investigated a relationship between QF-3 modules and localized Morita dualities. In this paper we shall introduce and investigate relative QF-3' modules and modules. Relative QF-3' modules are generalizations of QF-3'

We introduce for any Grothendieck category the notion of stable localizing subcategory, as a localizing subcategory that can be written as an intersection of localizing subcategories defined by indecomposable injectives. A Grothendieck category in which every localizing subcategory is stable is called a locally stable category. As a main result we give a characterization of these categories in terms of the local stability of a localizing subcategory and its quotient category. The locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally noetherian categories are examples of locally stable categories. We also present some applications to the category of modules over a left fully bounded noetherian ring, to the category of comodules over a coalgebra and to the category of modules over graded rings.

This note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q g of Q by g, for every g ∈ G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G r (r ≥ 1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ⊕ n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.

A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a direct summand of M. We prove that if M is a coatomic cofinitely Rad-⊕-supplemented module, then M is an irredundant sum of local direct summands. We show that the classes of cofinitely Rad-⊕-supplemented modules and Rad-⊕-supplemented modules are closed under finite direct sums. We also show that every direct summand of a weak duo (cofinitely) Rad-⊕-supplemented module is (cofinitely) Rad-⊕-supplemented.

This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

We study the t t -structure induced by an n n -tilting module T T in the derived category D ( R ) \mathcal {D}(R) of a ring R R . Our main objective is to determine when the heart of the t t -structure is a Grothendieck category. We obtain characterizations in terms of properties of the module category over the endomorphism ring of T T and, as a main result, we prove that the heart is a Grothendieck category if and only if T T is a pure projective R R -module.

Classes of operators determined by ordinal indices

We study transfer of pure-direct-objects via functors between Grothendieck categories. We consider fully faithful functors and adjoint pairs of functors. We give applications to module categories and comodule categories.

A ring $R$ is said to have a left Morita duality with a ring $S$ if there is an additive contravariant equivalence between two categories of left $R$-modules and right $S$-modules which include all finitely generated modules in $_R\mathfrak {M}$ and ${\mathfrak {M}_S}$ respectively and which are both closed under submodules and homomorphic images. We show that for such a ring $R$ the endomorphism ring of every finitely generated projective left $R$-module $_RP$ has a left Morita duality with the endomorphism ring of a suitably chosen cofinitely generated injective left $R$-module $_RQ$. Specialized to injective cogenerator rings and quasi-Frobenius rings our results yield results of R. L. Wagoner and Rosenberg and Zelinsky giving conditions when the endomorphism ring of a finitely generated projective left module over an injective cogenerator (quasi-Frobenius) ring is an injective cogenerator (quasi-Frobenius) ring.

We introduce the category of bicomodules for a comonad on a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [14], and the characterization of coseparable coalgebra coextensions stated in [19].

So far our emphasis has been on studying rings in terms of the module categories they admit—that is, in terms of the representations of the rings as endomorphism rings of abelian groups. As we shall see the Wedderburn Theorem for simple artinian rings can be interpreted as asserting that a ring R is simple artinian if and only if the category R M is “the same” as the category D M for some division ring D. On the other hand, if D is a division ring, then the theory of duality from elementary linear algebra asserts that the categories D FM and FM D of finitely generated left D-vector spaces and right D-vector spaces are “duals” of one another.