Endoprimitive rings

A class of endoprimitive rings is defined which contains all strongly prime rings and all weakly primitive rings. These rings are characterized in several ways and various facts about strongly prime and weakly primitive rings are generalized. An analog of the Jacobson radical is defined which is shown to be a normal radical lying between the Jacobson and Levitzki radicals.     

关于内射模的P-平坦性

通过引用P-平坦模的定义,引入了右IPF环的概念,推广了右IF环的概念,这对研究IF环及QF环具有重要的作用,同时对右IPF环的性质作了一些刻画,得到了右IPF环的若干个等价命题;最后,用P-平坦模及右IPF环推出了正则环的一些等价条件．     

Finite conductor rings

We extend the definition of a finite conductor domain to rings with zero divisors, and develop a theory of these rings which allows us, among other things, to provide examples of non-coherent finite conductor domains, and to clarify the behavior of non-coherent polynomial rings, group rings and fixed rings over coherent rings.

     

Weakly semisimple modulesand density theory

The objective of this article is to describe the theory of a class of semiprime rings which bear the same relationship to weakly primitive rings that semiprimitive rings have to primitive rings. In this theory, the role of semisimple modules is assumed by modules which are locally finite dimen- sional, polyform and weakly compressible.     

J-clean环的推广

如果R中每个元素(对应地,可逆元)均可表示为一个幂等元与环R的Jacobson根中一个元素之和,则称环R是J-clean环(对应地,UJ环).所有的J-clean环都是UJ环.作为UJ环的真推广,本文引入GUJ环的概念,研究GUJ环的基本性质和应用.进一步地,研究每个元素均可表示为一个幂等元与一个方幂属于环的Jacobson根的元素之和的环.     

The Jacobson Property of Group Rings and Its Generalizations

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Rings whose nilpotents form a multiplicative set

Nilpotents in Armendariz and abelian π-regular rings are multiplicatively closed. However, this fact need not hold in many kinds of rings. This article concerns a class of rings whose nilpotents are closed under multiplication. Rings contained in this class are said to be nilpotent-closed, and the structure of nilpotent-closed rings is investigated in relation with various situations which happen ordinarily in the study of noncommutative ring theory. In the procedure, various sorts of rings are investigated, so that they are nilpotent-closed. We also study familiar conditions under which nilpotents in nilpotent-closed rings form a subring.     

The absolutely Koszul property of Veronese subrings and Segre products

Absolutely Koszul algebras are a class of rings over which any finite graded module has a rational Poincaré series. We provide a criterion to detect non-absolutely Koszul rings. Combining the criterion with Macaulay2 computations, we identify large families of Veronese subrings and Segre products of polynomial rings which are not absolutely Koszul. In particular, we classify completely the absolutely Koszul algebras among Segre products of polynomial rings, when the base field has characteristic 0.     

关于半群环与群环的链条件的等价性

给出了将半群环的链条件转化为群环的链条件的一个定理,并由此将［１］中的结果推广到半群环的情形．     

Local rings whose modules are almost serial

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R-modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences.     

COMMUTATIVE ALTERNATIVE RINGS: A CONSTRUCTION

In this paper, the authors establish a connection between commutative, alternative rings and Engel rings(of type 2), thereby providing a method for constructing commutative, alternative rings which are not associative.     

Right Bézout ring with waist is a right Hermite ring

We study noncommutative rings in which the Jacobson radical contains a completely prime ideal. It is proved that a right Bézout ring in which the Jacobson radical contains a completely prime ideal is a right Hermite ring. We describe a new class of Bézout rings that are not elementary divisor rings.     

On graded nil clean rings

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.     

关于有限生成投射模为自由模的环

若环R上一切有限生成投射模都是自由模,则称R为PF环,本文讨论了PF环的一些性质及其一些应用。     

When is Every Module with Essential Socle a Direct Sums of Quasi-Injectives?

In the present article we study the structure of rings, over which essential extensions of semisimple modules are direct sums of quasi-injectives. In the special case of commutative rings, these rings are precisely Artinian PIR and so every module over such rings is a direct sum of cyclics as characterized by Köthe and Cohen-Kaplansky.     

Doset Hibi rings with an application to invariant theory

We define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal a?ne semigroup rings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is F-rational otherwise. We also state criteria of Gorenstein property of these rings.     

The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.     

有限扩张环

证明了环的有限扩张性可以传递到矩阵环上;通过PP环,半遗传环以及有限余非奇异环刻划了有限扩张环,并推广了文献[2]的定理2.1; 对于FGF与CF猜测,给出了部分肯定的回答,即右有限扩张右CF环是右CEP的,从而是右aritian的,改进了文献[6]的定理3.7.     

Reflexivity with maximal ideal axes

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. We in this note study rings with the reflexivity whose axis is given by maximal ideals (simply, an RM ring) which are a generalization of symmetric rings. It is first shown that the reflexivity of a ring and the RM ring property are independent of each other, noting that both of them are generalizations of ideal-symmetric rings. We connect RM rings with reflexive rings in various situations raised naturally in the procedure. As a generalization of RM rings, we also study the structure of the reflexivity with the maximal ideal axis on idempotents (simply, an RMI ring) and then investigate the structure of minimal non-Abelian RMI rings (with or without identity) up to isomorphism.     

Trivializable rings of sets

In the present paper, we examine the ways in which rings of sets can be generated by as simple systems as possible. We investigate rings of sets which can be generated by totally ordered subsystems. All totally ordered systems which generate the same ring are, in essence, isomorphic. This fact leads to an order-theoretic classification of rings of sets. Every group-valued mapping defined on a totally ordered system of sets can be uniquely extended to an additive mapping on the ring of sets generated by the system. Two rings of sets are of the same order type if and only if the corresponding spaces of measures are isomorphic.