关于l－群的半单结构

令Ｇ是一个ｌ－群，Ｇ的一个凸ｌ－子群Ａ叫做多余的，如果对Ｇ的任－凸ｌ－子群Ｗ，只要Ａ∨Ｗ＝Ｇ，就有Ｗ＝Ｇ．复令Ｒ（Ｇ）为Ｇ的所有多余凸ｌ－子群的集合并生成的凸ｌ－子群．我们证明了Ｒ（Ｇ）是ｌ－群Ｇ的一种报并且是在Ａｍｉｔｓｕｒ－Ｋｕｒｏｓｈ意义下的根．进一步我们得到了有限值ｌ－群的半单结构定理即Ｒ（Ｇ）＝０当且仅当Ｇｌ－同构于具有半单性的单ｌ－群的亚直积，同时我们还得到了一系列有意义的推论．     

Schur—Zassenhaus定理的推广

本文证明了如下结论：（１）若有限群Ｇ的一个Ｈａｌｌπ－子群Ｈ在ＧＦ内是Ｓ－半正规的，则Ｈ在Ｇ内有补且所有这样的补在Ｇ中互相共轭，（２）令Ｐ／Ｇ／，若有限群Ｇ的Ｓｙｌｏｗｐ－子群在Ｇ内是Ｓ－半正规的，则Ｇ是ｐ－可解的；（３）如果Ｇ与ＰＳＬ（２，７）是无关的，则Ｇ是π－可分的；（４）令Ｐ是一个奇素数，则其每个极小Ｐ－子群一Ｓ－半正规的有限群Ｇ－一定是Ｐ＝超可解的。     

格序群的C-根类和K-根类

本文研究了Ｃ-根类（即凸ｌ－子群格可分辨根类）和Ｋ－根类．证明了：投射ｌ－群、强投射ｌ－群、两两分裂ｌ－群、Ｆ－群等许多ｌ－群类都是ｌ－子群格可分辨的,但Ｈａｍｉｌｔｏｎｌ－群不是;Ｃ－根类格是根类格的一个有伪补的闭子格,而且是根类半群的一个子半群．每个凸Ｏ一子群都是闭的,而且给出了由一簇Ｋ一根类及一个根类所生成的Ｋ－根类的形式．     

有限交换环上极限线性群的Sylowp—子群

本文给出了有限交换局部环Ｒ上无限线性群ＧＬ（Ｒ）＝∪ｎＧＬｎＲ的Ｓｙｌｏｗｐ－子群的形式．令Ｍ是有限交换局部环Ｒ的唯一极大理想，ｋ＝Ｒ／Ｍ为Ｒ的剩余类域．用Ｘ（ｋ）表示ｋ的特征，并假定Ｐ与ｘ（ｋ）互素．作者证明了：ＧＬ（Ｒ）的任一Ｓｙｌｏｗｐ－子群Ｓ或者同构于的可数无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ≠２或Ｐ＝２，Ｘ（ｋ）β≡１（ｍｏｄ４））或者同构于Ｐｉ的无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ＝２，Ｘ（ｋ）β≡３（ｍｏｄ４）），这里，只是ＧＬ（ｅｐｉ）Ｒ（分别地，ＧＬ（２ｒｉ）Ｒ）的Ｓｙｌｏｗｐ－子群，Ｐ（ｊ））同构于Ｐ＝∪ｉ∈Ｉｐｉ，Ｉ是可数集．     

分次Morita对偶，Morita对偶与Smash积

设Ｃ和ｒ都是群，是Ｇ－型分次环，是Γ－型分次环．是双分次模，Ｒ＃Ｇ是Ｒ的Ｓｍａｓｈ积，Ａ＃Γ是Ａ的Ｓｍａｓｈ积。令Ｗ＝（＿ｇＵ＿（σ－１））＿（ｇ，σ）即（ｇ，σ）位置取＿ｇＵ＿（σ－１）的元素的｜Ｇ｜×｜Γ｜矩阵的全体组成的集合，且每个矩阵的每行和每列的非零元只有有限个，按矩阵运算，Ｗ构成（Ｒ＃６，Ａ＃Γ）双模。则＿ＲＵ＿Ａ定义了一个分次Ｍｏｒｉｔａ对偶当且仅当＿（Ｒ＃Ｇ）Ｗ＿（Ａ＃Γ）定义了一个Ｍｏｒｉｔａ对偶。     

同无限二面体群关联的晶体群的分类

设Ｖ是实数域Ｒ｝上的一个２－维向量空间，Ｖ带有一个仿射的或不定的对称双线性型．无限二面体群Ｗ能够被看作ＧＬ（Ｖ）的一个子群．在本文中，在仿射群Ａ（Ｖ）中共轭的意义下将分类同Ｗ关联的所有晶体群．     

L_2（q）的一个特征性质

设Ｇ是有限群，πｓ（Ｇ）为Ｇ的极大子群阶之集．本文证明了若ｑ＝ｐｎ＞２，ｐ素，则Ｇ≌Ｌ２（ｑ）当且仅当πｓ（Ｇ）＝πｓ（Ｌ２（ｑ））．对一些其它的单群也证明了同样的结论．     

有限Abel群的一个特征

本文证明了有限群Ｇ是Ａｂｅｌ群当且仅当Ｇ_ｒ满足下列条件：（Ⅰ） Ｇ有一个幂自同构 ａ使得 ＣＧ（ａ）是一个初等 ＡｂｅｌＺ一群．（Ⅱ）Ｇ没有子群与２－群＜ａ,ｂ｜ａ~２~ｎ＝ｂ~２~ｍ＝１,ａ~ｂ＝ａ~（１＋２）~（ｎ－１）＞同构,其中ｎ≥３,ｎ≥ｍ．利用该结果,作者还证明若有限群Ｇ有一个幂自同构ａ使得Ｃ_Ｇ（ａ）是一个初等Ａｂｅｌ２－群,则Ｇ是幂零群     

群环上的幂自由模

本文证明了：设Ｒ为ｃｈａｒＲ≠０，Ｇ为有限生成的Ａｂｅｌ群，则：Ｐ∈Ｆ＝（ＲＧ）当且仅当ｓ＞０，使得Ｐ^ｓ∈Ｆ＝（ＲＧ）．     

有限交换环上线性群的Carter子群

令Ｒ为有限交换局部环，Ｋ为其剩余类域．本文研究了Ｒ上一般线性群ＧＬｎＲ的Ｃａｒｔｅｒ子群的存在性及结构．得到的结果是：若ｃｈａｒＫ为奇数或Ｋ＝Ｆ２，ＧＬｎＲ中存在唯一的Ｃａｒｔｅｒ子群的共轭类，即Ｓｙｌｏｗ－２子群的正规化子；若ｃｈａｒＫ＝２且｜Ｋ｜＞２，ＧＬｎＲ中不含Ｃａｒｔｅｒ子群．     

Soluble Groups and Varieties of <Emphasis Type="Italic">l</Emphasis>-Groups

A sufficient condition is given under which factors of a system of normal convex subgroups of a linearly ordered (l.o.) group are Abelian. Also, a sufficient condition is specified subject to which factors of a system of normal convex subgroups of an l.o. group are contained in a group variety . In particular, for every soluble l.o. group G of solubility index n, n ⩾ 2, factors of a system of normal convex subgroups are soluble l.o. groups of solubility index at most n − 1. It is proved that the variety of all lattice-ordered groups, approximable by linearly ordered groups, does not coincide with a variety generated by all soluble l.o. groups. It is shown that if is any o-approximable variety of l-groups, and if every identity in the group signature is not identically true in , then contains free l.o. groups.Supported by FP “Universities of Russia” grant UR. 04. 01. 001.__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 355–367, May–June, 2005.     

Categories of semigroups in quantum computational structures

We investigate a categorial duality between quasi MV-algebras (a variety of algebras arising from quantum computation and tightly connected with fuzzy logic) and a reflective subcategory of l-groups with strong units.      

A Note on Supplementing Radicals

ＡＮｏｔｅｏｎＳｕｐｐｌｅｍｅｎｔｉｎｇＲａｄｉｃａｌｓＲｉｃｈａｒｄＷｉｅｇａｎｄｔ（ＭａｔｈｅｍａｔｉｃａｌＩｎｓｔｉｔｕｔｅ，ＨｕｎｇａｒｉａｎＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，Ｐ．Ｏ．Ｂｏｘ１２７，Ｂｕｄａｐｅｓｔ，Ｈｕｎｇａｒｙ）Ａｂｓｔ...     

A Note on Rings and Modules Graded by Some Special Groups

In this note we prove that the prime madicals of some graded rings and the scoles of some graded modules are also graded. The results on the relations between the prime radical and the graded prime radical are also presented. Moreover, the relations between a graded ring and its initial subring are studied.     

THE M-VALUATION SPECTRUM OF A COMMUTATIVE RING

Abstract

In 1945, N. Jacobson has introduced the definition of radical of a ring A (which is known as “Jacobson radical”, and is denoted J = J(A)). Later the concept of (Jacobson) radical of a left (or right) A-module M, J(M), has been defined as the intersection of all submodules N ≤ M such that M/N is simple. Thus one may consider the left radical J _l  = J( _A A) and the right radical J _r  = J(A _A ) of A, which are bilateral ideals of A, and are contained in J(A). If A has identity, one has J = J _l  = J _r , but this equality is not valid in general. Dual, it is possible to define left socle S _l and right socle S _r of A. We shall establish relations between J, J _l , J _r , S _l and S _r , and for artinian algebras we shall obtain expressions for J _l (A) and J _r (A), S _l (A) and S _r (A). In particular, if A is a finite dimensional algebra over a field we display J _l  = J( _A A) (and J _r  ? J(A _A )) in a matrix representation.     

The Hopf-Reflected Radicals for Hopf Module Algebras

０．ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ，ＨａｌｗａｙｓｄｅｎｏｔｅａＨｏｐｆａｌｇｅｂｒａｏｖｅｒａｆｉｅｌｄＫ，ａｎｄｗｅｆｏｌｏｗｔｈｅｎｏｔａ－ｔｉｏｎｓｉｎＳｗｅｅｄｌｅ’ｓｂｏｏｋ［１］．ＦｏｒｇｅｎｅｒａｌｔｈｅｏｒｙｏｆＨ－...     

关于绝对半素环和绝对半素根

本文了绝对半素环的概念，研究了绝对半素与与半素环及素环之间的关系，讨论了由全体绝对半素环类决定的上根－－绝对半半素根及其性质。     

Saturations of Submodules

Abstract

Let N be a submodule of a module M over a commutative ring R such that M/N is finitely generated. It is shown that a submodule of M is a prime submodule minimal over N if and only if it is the saturation of N + pM for certain prime ideal p of R. The bearing of this result upon the M-radical of N is discussed.