On normal classes of rings

In [6] Lanski,Resco and Small proved that, if I is a non-zero right ideal of a prime ring R then R is right primitive if and only if I is right primitive modulo its prime radical. Considering the opposite ring one gets the left version of this result. It is natural to ask whether the mixed version C left ideal, right primitivity) of the theorem holds as well. Studying this question we concluded that all the results of [6] can be extended to normal classes of rings[7] (of which the classes of left and right primitive rings are examples). It in particular answers positively the question. we also get several new characterizations of normal classes and find a direct proof of the quoted result of [6].     

Two Dimensional Nilpotent Euclidean Nearrings

We find all multiplications on the two dimensional Euclidean group (IR²,+) such that (IR², +, ·) is a nilpotent topological nearring of rank three. We determine the left, right, and two sided ideals of these nearrings and we determine when two of these nearrings are isomorphic. As a consequence, we are able to conclude that there are infinitely many isomorphism classes of such nearrings. This is in considerable contrast to the case for rings. There are only eight isomorphism classes of topological rings with additive group (IR²,+).AMS Subject Classification (2000) 16Y30     

The Krull-Schmidt Theorem in the Case Two

We study what happens if, in the Krull-Schmidt Theorem, instead of considering modules whose endomorphism rings have one maximal ideal, we consider modules whose endomorphism rings have two maximal ideals. If a ring has exactly two maximal right ideals, then the two maximal right ideals are necessarily two-sided. We call such a ring of type 2. The behavior of direct sums of finitely many modules whose endomorphism rings have type 2 is completely described by a graph whose connected components are either complete graphs or complete bipartite graphs. The vertices of the graphs are ideals in a suitable full subcategory of Mod-R. The edges are isomorphism classes of modules. The complete bipartite graphs give rise to a behavior described by a Weak Krull-Schmidt Theorem. Such a behavior had been previously studied for the classes of uniserial modules, biuniform modules, cyclically presented modules over a local ring, kernels of morphisms between indecomposable injective modules, and couniformly presented modules. All these modules have endomorphism rings that are either local or of type 2. Here we present a general theory that includes all these cases.     

Rings whose singular ideals are nil

Abstract

It is well known that when a ring R satisfies ACC on right annihilators of elements, then the right singular ideal of R is nil, in this case, we say R is right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial rings and triangular matrix rings. The class of right nil-singular rings contains π-regular rings and is closed under direct sums. Examples are provided to explain and delimit our results.     

Preinjective modules and finite representation type of artinian rings

Let R be a left artinian ring such that every finitely presented right .ft-module is of finite endolength. It is shown that the cardinality of the set of isomorphism classes of preinjective right R-modules is less than or equal to the cardinality of the set of isomorphism classes of preprojective left R-modules, and R is of finite representation type if and only if these cardinal numbers are finite and equal to each other. As a consequence, we deduce a theorem, due to Herzog [17], asserting that a left pure semisimple ring R is of finite representation type if and only if the number of non-isomorphic preinjective right R-modules is the same as the number of non-isomorphic preprojective left .R-modules. Further applications are also given to provide new criteria for artinian rings with self-duality and artinian Pi-rings to be of finite representation type, which imply in particular the validity of the pure semisimple conjecture for these classes of rings.     

TORSION PRERADICALS—A TOOL FOR ANALYSING RINGS

Abstract

This paper surveys a selection of results in the literature on torsion preradicals; these are left exact preradical functors on the category of unital right modules over an associative ring with identity. Various well known classes of rings such as semisimple, artinian, perfect and strongly prime are characterized in terms of torsion preradicals. A classification of prime rings using torsion preradicals is also exhibited. Rings all of whose torsion preradicals are radicals and rings whose torsion preradicals commute, are investigated. An application of the latter condition to Jacobson's Conjecture is presented.     

Some Classes of Nilpotent Associative Algebras

We further the study of rings with no middle class by focusing on an interpretation of that property in terms of the lattice of hereditary pretorsion classes over a given ring. For non-semisimple rings, the absence of a middle class is equivalent to the requirement that the class of all semisimple right modules be a coatom in that lattice. Taking advantage of this perspective, we discover new facts and shed light on others already known with a possibly more direct interpretation without having to refer to an exhaustive analysis of the structure theorems available in the literature. Our approach also allows us to characterize rings with no middle class in terms of hereditary pretorsion classes containing the class of all singular right modules. We discuss the open problem of whether there is a ring with no right middle class which is not right Noetherian and see, in particular, that an indecomposable ring satisfying that property would have to be Morita equivalent to a certain type of subring of a full linear ring.     

MP-injective rings and MGP-injective rings

A ring R is said to be right MP-injective if every monomorphism from a principal right ideal to R extends to an endomorphism of R. A ring R is said to be right MGP-injective if, for any 0 ≠ a ∈ R, there exists a positive integer n such that a ⁿ ≠ 0 and every monomorphism from a ⁿ R to R extends to R. We shall study characterizations and properties of these two classes of rings. Some interesting results on these rings are obtained. In particular, conditions under which right MGP-injective rings are semisimple artinian rings, von Neumann regular rings, and QF-rings are given.     

关于m-fold稳定环

本文给出了R为m-fold稳定环的若干充分必要条件，证明了整闭整环的Kronecker函数环m-fold稳定环，进一步地，得到了左(右)拟DUO替换环为m-fold稳定环的条件。     

On ℵ0-Injectivity

Abstract

In this note we investigate ?₀-injectivity of rings and modules and review the literature around this topic. We observe that several characterizations of rings by injectivity can be expressed by ?₀-injectivity. Moreover we point out that the following three classes of rings are not axiomatizable: the right ?₀-self-injective rings, the right ?₀-self-injective regular rings and the regular Baer rings.     

Weakly Special Classes of Semiprime Rings

This paper investigates closure properties possessed by certain classes of finite subdirect products of prime rings. If ℳ is a special class of prime rings then the class ℳ_∞ of all finite subdirect products of rings in ℳ is shown to be weakly special. A ring S is said to be a right tight extension [resp. tight extension] of a subring R if every nonzero right ideal [resp. right ideal and left ideal] of S meets R nontrivially. Every hereditary class of semiprime rings closed under tight extensions is weakly special. Each of the following conditions imposed on a semiprime ring yields a hereditary class closed under right tight extensions: ACC on right annihilators; finite right Goldie dimension; right Goldie. The class of all finite subdirect products of uniformly strongly prime rings is shown to be closed under tight extensions, answering a published question. This revised version was published online in June 2006 with corrections to the Cover Date.     

ON PRIME RINGS AND THE RADICALS ASSOCIATED WITH THEIR DEGREES OF PRIMENESS

Abstract

Prime ringsMaybe classified by the sizes of the sets that ‘insulate’ their elements from annihilation. For a cardinal m > 0, the class [Pbar]_r,(m) of all rings that are right prime of ‘bound at most m’ is studied, with particular reference to its closure under constructions such as matrix rings, semigoup rings, orders and extensions. The classes [Pbar]_r,(m) are special in the sense of radical theory for each m > 0. The attendant upper radicals υ[Pbar]_r,(m) are right (and not left) strong; their compatibility with certain ring constructions is examined. In the lattice of radicals (where they form a strictly descending chain), their positions are described, relative to various familiar radicals.     

Injective modules and representational repleteness

The purpose of this paper is to investigate the relationship between the associated primes of non-zero factors of a uniform injective module over a Noetherian ring and the right clique of the assassinator, when the clique satisfies the second layer condition. We introduce the terminology “representationally replete” to refer to primes where every linked prime occurs as such an associated prime and we study this property in several classes of rings including enveloping algebras of solvable Lie algebras and finite dimensional algebras     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

Asymmetry in the Converse of Schur's Lemma

We show that the converse of Schur's Lemma can hold in the category of right modules, but not the category of left modules, over an appropriate ring. We exhibit classes of rings over which this left-right asymmetry does not occur, and provide new constructions of rings over whose module categories the converse of Schur's Lemma holds. We propose various open problems and avenues for further research concomitant to our work.     

STRONGLY PRIME GROUP RINGS

Abstract

A ring R is (right) strongly prime (SP) if every nonzero two sided ideal contains a finite set whose right annihilator is zero, SP rings have been studied by Handelman and Lawrence who raised the problem of characterizing SP group algebras. They showed that if R is SP and G is torsion free Abelian, then the group ring RG is SP. The aim of this note is to determine some more group rings which are SP.

For a ring R we also define the strongly prime radical s(R). We then show that s(R)G = s(W) for certain classes of groups.     

The topological jacobson radical of rings. II

In this paper, topologically primitive rings and rings possessing a faithful topologically irreducible module and bounded by this module are considered for the investigation of properties of their topological Jacobson radical. We investigate the topological Jacobson radical in some classes of topological rings such as left topologically Artinian rings, topological rings possessing a basis of neighborhoods of zero consisting of ideals, compact rings, and bounded strictly linearly compact rings.     

On Weak Dual Rickart Modules and Dual Baer Modules

We introduce and study the notion of wd-Rickart modules (i.e. modules M such that for every nonzero endomorphism ? of M, the image of ? contains a nonzero direct summand of M). We show that the class of rings R for which every right R-module is wd-Rickart is exactly that of right semi-artinian right V-rings. We prove that a module M is dual Baer if and only if M is wd-Rickart and M has the strong summand sum property. Several structure results for some classes of wd-Rickart modules and dual Baer modules are provided. Some relevant counterexamples are indicated.     

Gorenstein Projective Precovers

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.