亚直不可约环是体的条件

设R是结合环,H是R中全部非零理想之交,若H≠(0),则称R是亚直不可约环。本文研究了亚直不可约环是体的条件,得到: 定理 R是具有极小单侧理想的亚直不可约环,且H中无非零幂零元,则R是体。     

亚直不可约环成为全矩阵环的条件

给出了亚直不可约环是除环上的全矩阵环的一个新的条件。     

半质环的交换性定理

任意半质PI－环的交换性问题是环的交换性理论的一个主要内容,本文利用亚直不可约环和分式环的性质,研究了任意半质PI－环的交换性问题,证明了半质PI－环的交换性定理。     

关于伪准素环的注记

一个有单位元的交换环Ｒ称为伪准素环，如果Ｒ的每个非零理想都是某个准素理想之幂．本文证明了环Ｒ是伪准素环当且仅当Ｒ是准素环或Ｒ是两个域的直和或Ｒ是至多具有三个素理想的一维局部环，并且每个非零理想或是某个极小素理想之幂或是某个属于极大理想的准不比理想之幂.     

亚直不可约DQrC-环

在文[1]中我们引进了DQrC-环,本文给出了忠实的与不忠实的DQrC-环为亚直不可约环的充分必要条件.由于DQrC-环是比DQC-环更广泛的环类,所以本文推广了W.Burgess等人在DQC-环方面相应的工作.     

一类非结合泛代数的次直积分解

本文引入一类非结合泛代数，即零积结合近环，研究其次直积分解，得到两个结构定理。设Ｎ是零积结合分配生成近环，本文证明了：（ｉ）如果Ｎ是次直不可约的且无非零的二次幂零元，则Ｎ是整的；（ｉｉ）Ｎ是零积结合分配生成整近环的次直积当且仅当Ｎ不含非零的二次幂零元。这些结果在这一类泛代数中加强了著名的Ｂｉｒｋｈｏｆｆ定理。     

左零因子理想具升链条件之环是左Noether环

谢邦杰〔1〕中引入了“左零因子理想具升链条件”的环这一概念,继而证明了这种环的诣零单边理想是幂零的,本文证明这种环若含有非零零因子,它就是左 Noether 环,因此〔1〕中结论就是 Levitzki 定理〔2〕,本文采用〔1〕中的定义.定义 设 R 是一环,L 是 R 的一个左理想,如果 S 是 R 的一个子集合而由非零元素     

半素F—环的结构

如果环R含有一个有限非零元集X,使得任意非零aR与X之交不空,则称R为F-环。文献[1]证明了:无非零幂零元的F-环是有限个除环的直和。 本文的结果是:R是半素F-环,当且仅当R是有限个除环上的全阵环的直和。从而[1]的结论就是我们结果的一个推论。     

BCI-代数的一个结构定理

证明了下面的结构定理：每一个非零的周期BCI-代数是一些次直不可约BCI-代数的次直积．     

关于具升链条件的DQC环

W·Burgess 和 M·Chacron 在文献〔1〕中刻划了亚直不可约 DQC 环·所谓 DQC 环R,就是 R 的任何理想 I 均由 I∩Q 生成的,这里 Q 称为 R 的拟中心,即 Q={r∈R|对任何 s∈R,存在 s′、s″∈R 使得 rs=s′r 和 sr=rs″}.显然,交换环、有1之双环(单边理想均为双边理想之环)都是 DQC 环.本文给出了 DQC 环具理想升链条件的一个充分必要条件以及 krull 交定理在 DQC 环中的一个推广.如无特别说明,本文中的理想均指双边理想,R 表示 DQC 环,Q 表示 R 的拟中心,(a)表示由元素a∈R 生成的双边理想.根据拟中心 Q 的定义,我们有:对任何a∈Q,(a)={ar+ma|r∈R,m 是整数}={ra     

Homomorphic images of subdirectly irreducible rings

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly irreducible ring by its monolith as well as R does not need to be a homomorphic image of a finite subdirectly irreducible ring. We provide an analogous characterization also for varieties of rings with unity, for the quasiregular rings, for the rings with involution and for their subvarieties of commutative rings.     

Varieties of rings,where all subdirectly irreducible finite rings are Armendariz ones

In this paper we study varieties of rings, where all subdirectly irreducible finite rings are Armendariz. We also describe the locally finite varieties of Armendariz rings.     

Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. ?vr?ek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzmán. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2ⁿ such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.     

A Note on the Subdirectly Irreducible ckham Algebras

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Subdirectly irreducible algebras with hyperidentities of the variety of De Morgan algebras

The paper characterizes the class of subdirectly irreducible algebras satisfying hyperidentities of the variety of De Morgan algebras. Such algebras are called subdirectly irreducible De Morgan quasilattices. The suggested characterization is quite close to that of the classical case of subdirectly irreducible DeMorgan algebras.     

On uniform acts over semigroups

The paper is devoted to the investigation of uniform acts over semigroups perceived as an overclass of subdirectly irreducible acts. We establish conditions for a uniform act to be subdirectly irreducible. In particular, we prove that uniform acts with two zeros are subdirectly irreducible. Ultimately we investigate monoids which are uniform as right acts over themselves and we describe regular monoids with this property.     

分次环，A，Aｅ和Smash积A＃G*是亚直既约环的条件

讨论了分次环Ａ，Ａ_ｅ和Ａ＃Ｇ^*的相关性质，在Ａ是分次忠实时，Ａ＃Ｇ^*是亚直既约环当且仅当Ａ_ｅ是亚直既约环；在Ａ是分次非退化时，Ａ＃Ｇ^*是Ｇ亚直既约环当且仅当Ａ是分次亚直既约环．     

State morphism MV-algebras

We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras.     

Subdirectly irreducible medial quandles

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.