Rings Characterized by a Class of Modules

ABSTRACT

Let ? be a class of right R-modules. The notions of ?-injectivity and ?-flatness are used to investigate right ?-Noetherian rings, right ?-hereditary rings, and right ?-coherent rings. For an almost excellent extension S ≥ R, if either ring is the above-mentioned ring, so is the other. Specializing the class ?, some known results can be obtained and extended as corollaries.     

Stable subsets of modules and the existence of a unit in associative rings

We obtain criteria for the existence of a (left) unit in rings (arbitrary, Artinian, Noetherian, prime, and so on) that are based on the systematic study of properties of stable subsets of modules and their stabilizers that generalize the technique of idempotents. We study a class of quasiunitary rings that is a natural extension of classes of rings with unit and of von Neumann (weakly) regular rings, which inherits may properties of these classes. Some quasiunitary radicals of arbitrary rings are constructed, and the size of these radicals “measures the probability” of the existence of a unit. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 596–611, April, 1997. Translated by A. I. Shtern     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups

We investigate the simple modules for the sporadic simple Mathieu groups M₂₂, M₂₃ and M₂₄ as well as those of the automorphism group, the covering groups and the bicyclic extensions of M₂₂ in characteristics 2 and 3. We determine the vertices and sources as well as the Green correspondents of these simple modules. We also find two 3-blocks with elementary abelian defect groups of order 9 in these groups which are Morita equivalent to their Brauer correspondents.     

Strongly Irreducible Submodules of Modules

Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.     

Cotilting modules with finite left perpendicular categories

We characterize a cotilting module T such that the left perpendicular category ⊥ T is of finite type.     

半量子群u~(≥0)

设u~(≥0)表示一个固定单李代数的半量子群,给出了u~(≥0)的性质和表示.证明了Hopf代数u~(≥0)不是拟余交换的,因此左u~(≥0)-模范畴不是辫子monoidal范畴.在权模范畴W中,给出了所有单对象和投射对象.最后描述了所有单的Yetter-Drinfel'd u~(≥0)-权模.