Roles In Ring Theory Research Articles

Localization of triangulated categories and derived categories

The notion of quotient and localization of abelian categories by dense subcategories (i.e., Serre classes) was introduced by Gabriel, and plays an important role in ring theory [6, 131. The notion of triangulated categories was introduced by Grothendieck and developed by Verdier [9, 16) and is recently useful in representation theory [8,4, 143. The quotient of triangulated categories by epaisse subcategories is constructed in [16]. Both quotients were indicated by Grothendieck, and they resemble each other. In this paper, we will consider triangulated categories and derived categories from the point of view of localization of abelian categories. Verdier gave a condition which is equivalent to the one that a quotient functor has a right adjoint, and considered a relation between tpaisse subcategories [16]. We show that localization of triangulated categories is similarly defined, and have a relation between localizations and Cpaisse subcategories. Beilinson, Bernstein, and Deligne introduced the notion of t-structure which is similar to torsion theory in abelian categories [2]. We, in particular, consider a stable t-structure, which is an epaisse subcategory, and deal with a correspondence between localizations of triangulated categories and stable r-structures. And then recollement, in the sense of [2], is equivalent to bilocalization. Next, we show that quotient and localization of abelian categories induce quotient and localization of its derived categories. In Section 1, we recall standard notations and terminologies of quotient and localization of abelian categories. In Section 2, we define localization of triangulated categories, and consider a relation between localizations and stable r-structures (Theorem 2.6). In Section 3, we show that if A + A/C is a quotient of abelian categories, then D*(A) -+ D*(A/C) is a quotient of triangulated categories, where * = + , -, or b (Theorem 3.2). Moreover,

A characterization of rings with Krull dimension

Throughout this paper rings will mean associative rings with identity and all modules are assumed to be unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, see for example [ 1, 2, 9-111. One of the most important results in this direction is the result of Osofsky [S, Theorem] which says that a ring R is semisimple (i.e., R is right Artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. Starting from this and in connection with a result of Vamos [ 1 l] the following result has been recently obtained in [2, Theorem 1.11: A ring R is right Artinian if and only if every cyclic right R-module is a direct sum of an injective module and a finitely cogenerated module. In the present paper we follow this investigation and aim to show a similar result for rings with Krull dimension. We consider the following condition about a ring R:

A characterization of artinian rings

Throughout this paper we consider associative rings with identity and assume that all modules are unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, even by their simple modules. See, for example, [2], [3], [6], [7], [13], [14], [15], [16], [18], [21]. One of the most important results in this direction is the result of Osofsky [14, Theorem] which says: a ring R is semisimple (i.e. right artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. The other one is due to Vamos [18]: a ring R is right artinian if and only if every cyclic right R-module is finitely embedded.