Artinian Rings Characterized by Direct Sums of CS Modules

Abstract

In this note, we show that the following are equivalent for a ring R for which the socle or the injective hull of R _R is finitely generated: (i) The direct sum of any two CS right R-modules is again CS; (ii) R is right Artinian and every uniform right R-module has composition length at most two. Next we give partial answers to a question of Huynh whether a right countably Σ-CS ring which either is semilocal or has finite Goldie dimension is right Σ-CS. We give characterizations, in terms of radicals, of when such rings are right Σ-CS. In particular, for the semilocal case, Huynh's question is reduced to whether rad(Z ₂(R _R )) is Σ-CS or Noetherian, where Z ₂(R _R ) is the second singular right ideal of R. Our results yield new characterizations of QF-rings.     

Artin模上的替代结构

该文得到了Ａｒｔｉｎ模的一个替代结构，并进而证明了紧Ｈａｕｓｄｏｒｆｆ空间上连续函数环具有该替代结构．     

Artin半单环的特征

本文证明了如下结果：环Ｒ是Ａｒｔｉｎ半单的当且仅当存在一个基数ｃ，使得任意左Ｒ－模是一个连续模和一个ｃ－限制的ＥＳ－模的直和，也当且仅当存在一个基数ｃ，使得任意左界Ｒ－模是一个拟投射模和一个ｃ－限制的ＥＳ－模的直和。     

Anisotropic Modules over Artinian Principal Ideal Rings

Let V be a finite-dimensional vector space over a field k, and let W be a 1-dimensional k-vector space. Let ?,?: V × V → W be a symmetric bilinear form. Then ?,? is called anisotropic if for all nonzero v ∈ V we have ? v, v ? ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that a form is anisotropic if and only if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss the radical root of a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules

In this article, considering the difference between the finiteness dimension and cohomological dimension of a finitely generated graded module, we investigate the asymptotic behaviour of grades of components of a graded local cohomology module with respect to the irrelevant ideal. To this end, we study some Artinian and tameness properties of certain graded local cohomology modules.     

The Theory of Local Homology for Artinian Modules

In this paper, the theory of local homology for Artinian modules is developed, which in some aspect is similar to the theory of local cohomology. There are given some criteria for local homology modules to be good and dual of the ideal transform functor is studied.AMS Subject Classification (1991): 13E10 13D45     

On Some Artinian QF-3 Rings

Carl Faith in 2003 introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings) [5 Faith , C. ( 2003 ). When cyclic modules have Σ-injective hulls . Comm. Algebra 13 : 4161 – 4173 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. Inspired by this we investigate rings over which every cyclic right R-module has a projective Σ-injective injective hull. We show that a ring R satisfies this condition if and only if R is right artinian, the injective hull of R _R is projective and every simple right R-module is embedded in R _R . We also characterize right artinian rings in terms of injective faithful right ideals and right CSI-rings.     

Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.     

Some Results for Formal Local Cohomology Modules

Let （R, m） be a commutative Noetherian local ring, I an ideal of R and M a finitely generated R-module. Let limnHm^i（M/I^nM）be the ith formal local cohomology module of M with respect to I.In this paper, we discuss some properties of formal local cohomology modules limnHm^i（M/I^nM）,which are analogous to the finiteness and Artinianness of local cohomology modules of a finitely generated module.     

On the Attached Prime Ideals of Local Cohomology Modules

In this article, we shall prove some new properties about attached prime ideals over local cohomology modules. Also we generalize some of the results of [2 Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press .[Crossref] , [Google Scholar]].     

On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action

Let be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring (associated to and the Frobenius homomorphism , in the indeterminate ) that is both -torsion and Artinian over .

     

关于G－Matlis自反模的几点注记

本文证明，如果Ｒ是一个Ｎｏｅｔｈｅｒ完备半局部环，则Ｒ－模Ｍ是Ｎｏｅｔｈｅｒ（Ａｒｔｉｎ）模当且仅当对任意ＡｒｔｉｎＲ－模Ｎ，Hom _R(M, N)是Ａｒｔｉｎ（Ｎｏｅｔｈｅｒ）模．