An approach to injective acts over monoids based on indecomposability

The main purpose of this paper is to investigate classes of acts that are injective relative to all embeddings with indecomposable domains or codomains. We give some homological classifications of monoids in light of such kinds of injectivity. Our approach to the indecomposability property provides a new characterization of right absolutely injective monoids such that all indecomposable acts are injective.     

Quasipure injective torsion-free groups with indecomposable pure subgroups

In a quasipure injective torsion-free Abelian group whose pure subgroups are strongly indecomposable, any nonzero endomorphism is shown to be a monomorphism. The results of this paper together with results obtained earlier describe quasipure injective torsion-free groups. As a corollary, it is proved that any quasipure injective torsion-free group is transitive. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 587–592, October, 2000.     

Main Injective Rings

We refer to those injective modules that determine every left exact preradical and that we called main injective modules in a preceding article, and we consider left main injective rings, which as left modules are main injective modules. We prove some properties of these rings, and we characterize QF-rings as those rings which are left and right main injective.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

子立方无爪图的单射边色数至多为6

设$G$是一个图. 图$G$的一个单射边染色是指图$G$的一个边染色, 使得距离为$2$的两条边或者在同一个三角形中的两条边染不同的颜色. 图$G$的单射边色数是指图$G$的任意单射边染色所需要的最少颜色数. 关于单射边色数有一个猜想: 任意一个子立方图的单射边色数都不超过$6$. 在本文中, 我们证明了这个猜想对子立方无爪图是成立的, 并且给出图例说明上界$6$是紧的. 同时, 我们的证明隐含了求解这类图不超过$6$种颜色的单射边染色方案的一个线性时间算法.     

使用相关内射性对拟连续模的一个刻划

近二十年，许多环与模工作对拟投射模与拟内射模作了各种推广与研究。连续模与拟连续模就是拟内射模的一种推广，拟连续模要比连续模弱。在[2]，作对连续模与拟连续模做了深入的研究。在这篇章中，利用相关内射性给出了拟连续模的一个刻划。     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

On Pseudo Small Injective Modules

The definition of principally pseudo injectivity motivates us to generalize tae notion of injectivity, noted SP pseudo injectivity. The aim of this paper is to investigate characterizations and properties of SP pseudo injective modules. Various results are devel- oped, many extending known results. As applications, we give some characterizations on Noetherian rings, QI rings, quasi-Frobenius rings.     

n- Gorenstein环上的 Gorenstein内射模(英文)

用 Gorenstein内射模刻画了 n-Gorenstein环 .     

On certain homological finiteness conditions

In this paper, we show that the injective dimension of all projective modules over a countable ring is bounded by the self-injective dimension of the ring. We also examine the extent to which the flat length of all injective modules is bounded by the flat length of an injective cogenerator. To that end, we study the relation between these finiteness conditions on the ring and certain properties of the (strict) Mittag–Le?er modules. We also examine the relation between the self-injective dimension of the integral group ring of a group and Ikenaga’s generalized (co-)homological dimension.     

弱直投射模和弱直内射模

引入了弱直投射和弱直内射模的概念，给出了它们的一些性质。使用弱直投射和弱直内射模刻划了遗传环、半遗传环、半单环和ＱＦ－环。     

Supplement Submodules of Injective Modules

An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R _R is F-almost injective.     

Characterizations of Strongly Regular Rings

ＣｈａｒａｃｔｅｒｉｚａｔｉｏｎｓｏｆＳｔｒｏｎｇｌｙＲｅｇｕｌａｒＲｉｎｇｓＺｈａｎｇＪｕｌｅ（章聚乐）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＡｎｈｕｉＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈｕ２４１０００）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉ...     

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

On finitely injective modules and locally pure-injective modules over Prüfer domains

Over Matlis valuation domains there exist finitely injective modules which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Prüfer domains which are either almost maximal, or -local Matlis, finitely injective torsion modules and complete torsion-free locally pure-injective modules correspond to each other under the Matlis equivalence. Almost maximal Prüfer domains are characterized by the property that every torsion-free complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind.

     

Gorenstein Dimensions over Ring Homomorphisms

In this paper, we study injective dimension and Gorenstein injective dimension over local ring homomorphisms. Some well-known results are generalized. For example, the Bass formula for Gorenstein injective dimension of complexes is extended. As applications, some characterizations of Gorenstein rings are obtained.     

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠ 0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.     

内射模的t－平坦性

设Ｒ为环，ｔ是左Ｒ－模范畴的一个遗传挠理论．文中证明了下述各点等价：（１）每个内射左Ｒ－模是ｔ－平坦的；（２）每个ｔ－有限表现左Ｒ－模的内射包络是ｔ－平坦的；（３）每个ｔ－有限表现左Ｒ－模是自由Ｒ－模的子模；（４）每个ｔ－有限表现左Ｒ－模是自反的且其对偶模是Ｈ－有限生成的．