Characterization of Hereditary Rings ViaInjectivity and Projectivity Classes

Throughout,allringsconsideredhaveanidentity,modulesareunitalleftmodulesandE(M)standsfortheinjectivehullofM.Now,letSbeaclassofmodules,thatisacollectionofR-modulessuchthatifM∈SthenanyR-moduleisomorphictoMbelongstoS.Wewillfreelyusethenotation,terminologyandresultsof[2].Recently,thereareanumberofwellknowntheoremswhichcharacterizeringsintermsofinjectiveandprojectivemodules(cf.[1],[3]and[4]).In[5,6]injectivityclassesandprojec-tivityclassesareintroducedandsimilarresultsremaintruewheninjectivity(…     

凝聚环和IF环

本文利用特征模，Ｎ０－内射维数对凝聚环给出了一些新的刻划．得到了凝聚环与ＩＦ环的一些有意义的性质．推广了引文［１］中的两个主要定理之一的定理２的结论．     

Caractérisation de certaines classes d'anneaux par des propriétés des endomorphismes de leurs modules

The aim of this paper is to study some classes of rings whose all modules have endomorphism rings with properties “similar” to those of the endomorphism rings of vector spaces.     

半正则环的推广

In this paper, the concept of generalized semiregular rings is extended to generalized weak semiregular rings. Some properties of these rings are studied and some results about semiregular rings and generalized semiregular rings are extended. We also give some equivalent characterizations ofI-weak semiregular rings.     

Quasi Co-Hopfian Modules and Applications

We carry out an extensive study of modules M _R with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective).     

Modules

A survey is given of results on modules over rings, covering 1976–1980 and continuing the series of surveys “Modules” in Itogi Nauki.     

On the Left Perpendicular Category of the Modules of Finite Projective Dimension

In this article, we characterize several properties of commutative noetherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application, we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.     

Countable Exchange and Full Exchange Rings

We show that a Dedekind-finite, semi-π-regular ring with a “nice” topology is an ?₀-exchange ring, and the same holds true for a strongly clean ring with a “nice” topology. We generalize the argument to show that a Dedekind-finite, semi-regular ring with a “nice” topology is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite exchange has countable exchange, and all modules with Dedekind-finite, semi-regular endomorphism rings are full exchange modules. These results are generalized further.     

幺半群的内射类及投射类刻画

类比于一般环上模的内射类,定义了幺半群上的S-系的内射类和投射类,并利用它们刻画了几类特殊的幺半群．证明了完全内射幺半群和完全拟内射幺半群是等价的．并且证明了对于标致幺半群S,它是完全投射的当且仅当它是完全拟投射的当且仅当它上面的投射S-系构成了一个投射类．     

Local Units Versus Local Projectivity Dualisations: Corings with Local Structure Maps

We unify and generalize different notions of local units and local projectivity. We investigate the connection between these properties by constructing elementary algebras from locally projective modules. Dual versions of these constructions are discussed, leading to corings with local comultiplications, corings with local counits, and rings with local multiplications.     

Minimalities and Modules over Dedekind-Like Rings

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings.     

Weakly dual automorphism invariant and superfluous cotightness

Abstract

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules.     

Relative Projectivity of the Modules $$L_p$$

Mathematical Notes - In the paper, criteria are given for the relative projectivity of the $$L_p$$ -spaces regarded as left Banach modules over the algebra of bounded measurable functions ( $$1\le...     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.     

Macros and modules in geometry

The paper highlights the importance of “macros” or modules for teaching and learning Geometry using Dynamical Geometry Software (DGS). The role of modules is analyzed in terms of “writing” and “reading” Geometry. At first, modules are taken as tools for geometrical construction tasks and as tools to describe and analyze these constructions. For proofs, decomposing a given geometrical statement may be supported by using prototypical pictures representing theorems of geometry (“modules”). Reading theorems into geometry and constructing proofs is still a major achievement of the student—which may be reached by using macros and modules as a major heuristic strategy     

Generalizations of projectivity and supplements revisited for superfluous ideals

We (re)introduce four ideal-related generalizations of classic module-theoretic notions: the ideal-superfluity, projective ideal-covers, the ideal-projectivity, and ideal-supplements. For a superfluous ideal I, the main theorem asserts the equivalence between the conditions: “I-supplements are direct summands in finitely generated projective modules”; “finitely generated I-projective modules are projective”; “projective modules with finitely generated factors modulo I are finitely generated”; “finitely generated flat modules with projective factors modulo I are projective.” Moreover, we provide a property of the ideal I which is sufficient for the equivalence to hold true. The property is expressed in terms of idempotent-lifting in matrix rings.     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

I半π正则环

Let R be a ring and I an ideal of R.A ring R is called I-semi-π-regular if R/I isπ-regular and idempotents of R can be strongly lifted modulo I.Charac- terizations of I-semi-π-regular rings are given and relations between semi-π-regular rings and semiregular rings are explored.