准正则环与正则环

每一个主左理想均由一个幂等元生成的环叫正则环。每一个左理想均由若干个幂等元生成的环叫准正则环。本文研究准正则环与正则环的一些性质，讨论准正则环成为正则环的一些条件，准正则环、正则环与Ｖ环之间的关系，准正则环成为Ａｂｅｌ正则环的条件。     

无限矩阵环和完备环

无限矩阵环和完备环郭善良（复旦大学）Ｓｈａｎｎｙ在１９７１证明了一个环Ｒ是半单Ａｒｔｉｎ环当且仅当Ｒ上的无限矩阵环是ＶｏｎＮｅｕｍａｎｎ正则环［１］。这也就是说一个环的无限矩阵环在一定程度上唯一确定了Ｒ本身。我们注意到若ＥｎｄＦ，为ＶｏｎＮｅｕｍｍａ...     

UQ-环和强UJII-环

本文引入了UQ-环和UJII-环的概念,推广了UJ-环.利用环论中元素的技巧,研究了UQ-环和UJII-环的性质和结构,相关结果丰富了环中关于元素分解的理论.     

Morphic环和G-morphic环的一些结果

讨论了morphic环,G-morphic环,PP环,GPP环,Bear环与正则环之间的关系.还证明了在约化环中,强正则环,正则环,π-正则环,G-π-正则环的等价性.     

一类环上HX环的结构

自李洪兴１９９１年提出了ＨＸ环以来，人们一直有这么一个问题没解决，就是是否存在非平凡的ＨＸ环的例子？但至今既没找到非平凡的ＨＸ环，也没有证明任一环Ｒ仅存在平凡的ＨＸ环。针对这个问题，本文提出并证明了一类环仅有平凡ＨＸ环，还给出了一系列的结构定理。这样，既为证明任一环Ｒ仅有平凡的ＨＸ环的猜想有新的启示，也为人们指明无须在这一类环上寻找非平凡ＨＸ环。     

矩阵环的半交换子环

称环R是半交换的,如果对任意a∈R,rR(a)是R的理想．若n≥2,则任意具有单位元的环R上的n阶上三角矩阵环不是半交换环．我们证明了reduced环上的上三角矩阵环的一类特殊子环是半交换环．     

Morphic-环与SF-环

以正则环为桥梁,研究了morphic-环与SF-环之间的关系.主要工作如下:(i)研究了SF-环成为morphic-环的若干条件;(ii)讨论了在一定条件下SF-环与morphic-环的等价性;(iii)给出了利用morphic-环对半单环在约化条件下的一个刻划.     

幂环

提出幂环概念,使环的提升更趋合理,建立了幂环的结构定理,并得到了某种条件下幂环同商环的关系。     

内同构环与内异环的结构

所有真子环都同构的结合环,称为内同构环,任两不同的子环都不同构的结合环,称为内异环.本文目的是给出内同构环与内异环的一些结构定理,从而基本上解决了Szasz F.A.提出的问题81:怎样的结合环,它的不同子环总不同构?     

交换半群环的分式环

设含幺交换环Ｒ对其乘法子集Ｔ的分式环为ＲＴ，交换幺半群Ｓ在其子半群∑处局部化为Ｓ∑本文证明了Ｒ［Ｓ］对于Ａ的分环式环Ｒ［Ｓ］ＡＭ 构于半群环ＲＴ［Ｓ∑］。     

Commutative neat group rings

A ring R is called clean if every element of it is a sum of an idempotent and a unit. A ring R is neat if every proper homomorphic image of R is clean. When R is a field, then a complete characterization has been obtained for a commutative group ring RG to be neat, but not clean. And if R is not a field, then necessary conditions are obtained for a commutative group ring RG to be neat, but not clean. A counterexample is given to show that these necessary conditions are not sufficient.     

On finitely embedded rings

A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case.     

Some minimal rings related to 2-primal rings

In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS I were given in order to show that the ring class inclusions were strict. Many of the rings given in the examples were infinite. In this paper, where possible, examples of minimal finite rings of the various types are given. Along with the rings in the previous taxonomy, NI, abelian and reflexive rings are also included.     

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.     

Minimal reversible nonsymmetric rings

Marks showed that ${\mathbb{F}}_2{Q}_8$, the ${\mathbb{F}}_2$ group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although ${\mathbb{F}}_2{Q}_8$ is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo.     

Polynomial extensions of quasi-Baer rings

Summary For a ring endomorphism &agr; and an &agr;-derivation &dgr;, we introduce &agr;-compatible rings which are a generalization of &agr;-rigid rings, and study on the relationship between the quasi Baerness and p.q.-Baer property of a ring R and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [6], [8] and [16].     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Commutative weakly nil-neat group rings

Abstract

Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results, namely those established by McGovern et al. in (J. Algebra Appl., 2015), by Danchev-McGovern in (J. Algebra, 2015) and by the present authors in (J. Math., Tokushima Univ., 2019), related to commutative nil-clean, weakly nil-clean and nil-neat group rings, respectively.     

Polynomial orbits in finite commutative rings

Let R be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.