C. Năstăsescu

Graded and filtered rings and modules

Graded and filtered rings and modules / , Graded and filtered rings and modules / , کتابخانه دیجیتالی دانشگاه علوم پزشکی ارومیه

Some constructions over graded rings: Applications

A well-known result of Dade [7; Theorem 2.81 states that R is a strongly graded ring o the functor ( ), is faithful o V,, = { 01, Va E G. After presenting in Section 0 some definitions and results about modules over graded rings, we construct in Section 1 the induced functor Ind: R,-mod --, R-gr and the coinduced functor Coind: R,-mod --f R-gr. The main result of this section is Theorem 1.1, where properties of these functors are given; we focus our attention on the study of the functor Coind. The Coind functor is then used for the construction of the gr-injective envelope of a graded module. We remark that the construction of the Coind functor appears in the proof of Theorem 2.1 of [ 141 and also in [ 111, where it is used for the study of Morita duality of graded rings. In Section 2, the graded rings of finite support are studied. The main results of this section are Theorem 2.1, where the gr-injective modules are studied (some consequences are also given), and Theorem 2.2 (a theorem of incomparability for prime ideals), which extends Theorem 3.3 of [6]. In Section 3, Dade’s result (Theorem 3.1) is extended using the notion of quotient category and the localizing subcategories ‘is,. Some applications of Theorem 3.1 are then given (Corollaries 3.1, 3.2, 3.3), extending some results of Menini and Nastasescu [12]. Finally, Theorem 3.2 is a generalization of Theorem 3.1, but the proof is of a different nature. 119 0021-8693/89

ON GRADINGS OF MATRIX ALGEBRAS AND DESCENT THEORY

3.00

Quasi-co-Frobenius Coalgebras. II

We classify gradings on matrix algebras by a finite abelian group. A grading is called good if all elementary matrices are homogeneous. For cyclic groups, all gradings on a matrix algebra over an algebraically closed field are good. We can count the number of good gradings by a cyclic group. Using descent theory, we classify non-good gradings on a matrix algebra that become good after a base extension.

Graded almost noetherian rings and applications to coalgebras

Abstract We prove new characterizations of Quasi-co-Frobenius (QcF) coalgebras and co-Frobenius coalgebras. Among them, we prove that a coalgebra is QcF if and only if C generates every left and every right C-comodule. We also prove that every QcF coalgebra is Morita-Takeuchi equivalent to a co-Frobenius coalgebra.

LOCALIZATION IN COALGEBRAS: APPLICATIONS TO FINITENESS CONDITIONS

It is well-known that the dual algebra of a coalgebra C is a topological algebra with the weak-∗ topology. In this paper we study some finiteness conditions relative to the topological structure of C∗ in terms of the category Rat(C∗M) of rational left C∗-modules. In particular, we focus on the problem whether Rat(C∗M) is closed under extensions. In torsion theoretic terms this raises the question of deciding when Rat(C∗M) is a torsion theory or a localizing subcategory in C∗M, the category of all left C∗-modules (the notion of localizing subcategory used here is as in [5], [19]). This problem has been previously treated in [9], [11], and [18], where a coalgebra satisfying this property is said to be a coalgebra having a torsion rat functor.

Relative Regular Objects in Categories

We introduce the notion of right strictly quasi-finite coalgebras, as coalgebras with the property that the class of quasi-finite right comodules is closed under factor comodules, and study its properties. A major tool in this study is the local techniques, in the sense of abstract localization.

Separable functors in graded rings

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the theory of generalized inverses of matrices is presented. Some of the techniques we use are new, since dealing with arbitrary categories allows us to pass to the dual category.

Relative graded Clifford theory

Abstract Separable functors were introduced by C. Năstăsescu et al. (J. Algebra 123 (1989) 397–413). We characterize separability of left or right adjoint functors defined on a Grothendieck category having a set of projective generators. This general results are particularized to the canonical functors arising from a graded homomorphism of group-graded rings (restriction of scalars, induction and coinduction functors). We relate the separability of these functors with that of their ungraded versions. In particular, we recover the characterizations given in loc. cited for the ungraded restriction of scalars and induction functors.

Infinite group-graded rings, rings of endomorphisms, and localization

We give a relative version of the ‘Graded Clifford Theorem’. The relative graded Clifford theorem is a powerful tool in the study of C-cocritical objects of the category R-gr where C is a rigid localizing subcategory of R-gr. We apply the result to the study of Gabriel (Krull) dimension of a graded module.