一般代数对象的挠理论

1993年Puczylowski在用公理系统构造的其中元素称为代数的对象类中建立了根与半单类的一般理论。本文的目的是在这种最具广泛性的代数系统中讨论挠理论，用格论方法给出它的一些特征刻划。     

可积代数正规类中半素代数类及半素一致代数类确定的上根

利用代数正规类中的理想乘积公理，引入可积代数正规类及可积代数正规类中素代数、半素代数类及一致代数类概念，讨论了可积代数正规类中半素代数类及半素一致代数类确定的上根性质。     

Pontrjagin空间上算子代数的分类和一般形式

给出Pontrjagin空间Ⅱ_k上一般算子代数的对称理想和非对称理想,利用这些理想给出算子代数的分类概念.将Ⅱ_k空间上一般算子代数分为六类,并给出各类充分大的(简称SM)代数的一般形式.     

三维半群代数

首先给出代数闭域上三维半群代数的幂等元集和Jacobson根,并且刻画了三维半群代数的同构类.通过计算箭图,研究了三维代数的表示型.进一步,证明一个三维(或者二维)半群代数是胞腔的,当且仅当它是交换的.作为推论,得到一个左零带所对应的半群代数是胞腔的,当且仅当这个左零带是一个半格.     

信息代数紧扩张的一种新方法

本文研究了从一般信息代数获取紧信息代数的一种新方法.通过在一个信息代数的定向集族上建立一个等价关系,证得相应的等价类全体之集关于联合和聚焦两种运算构成了原信息代数的一个紧扩张.     

某些完备李代数的结构

本文讨论了完备李代数的同构问题，并对幂零根基为一些Ｈｅｉｓｅｎｂｅｒｇ代数及交换李代数之直和的可解和一般完备李代数的结构进行了讨论．     

*-代数自由积上的线性泛函的延拓

本文研究了C*-代数及其*-稠子代数的*-代数自由积.利用自由积的性质,得到了这两类自由积上的线性泛函到C*-代数(泛)自由积上的态延拓的充要条件,从而证明了这类延拓对于一般的C*-代数也是成立的.     

Pontrjagin空间上J.V.N代数的导子

本文证明Pontrjagin空间上非退化的J.V.N代数的导子是内的等价于它在该代数的奇异部分上的限制为零.对退化的J.V.N代数,证明了Ⅱ1空间第0,Ⅱa和Ⅲa类J.V.N代数上的导子是内的.通过构造例子说明了第Ⅰ,Ⅱb和Ⅲb类对称代数上的导子一般不是内的.     

Pontrjagin空间上J.V.N代数的导子

本文证明Pontrjagin空间上非退化的J.V.N代数的导子是内的等价于它在该代数的奇异部分上 的限制为零.对退化的J.V.N代数,证明了Ⅱ1空间第0,Ⅱa和Ⅲa类J.V.N代数上的导子是内 的.通过构造例子说明了第Ⅰ,Ⅱb和Ⅲb类对称代数上的导子一般不是内的.     

一类推广的Virasoro-like李代数

构造了一类无限维李代数,它是Virasoro-like李代数的推广.研究了这类李代数的两类自同构,这两类自同构均关于映射的合成构成自同构群,一类同构于对称群S3,另一类同构于Klein交换群.得到了这类李代数一些特殊的自同态、中心.证明了这类李代数不是半单李代数.     

分次根与自反根

Ｍ．Ｂｅａｔｔｉｅ和Ｐ．Ｓｔｅｗａｒｔ在文（１）中讨论了六种分次根与相应自反根的关系。作为该文的继续，本文首先定义了Ｋｏｅｔｈｅ根并讨论它与Ｋｏｅｔｈｅ自反根的关系。其次，进一步给出Ｌｅｖｉｔｚｋｉ自反根的刻画。最后，我们指出文（１）的两个错误结论，证明分次ｖｏｎ Ｎｅｕｍａｎｎ根与ｖｏｎ Ｎｅｕｍａｎ自反根恰好相等。     

Radicals in semigroups

The general theories of radicals in rings and groups initiated by Amitsur [1] and Kurosh [6, 7] and developed by a number of authors in the past decade, were brought under a common roof by Hoehnke [3] in a theory of radicals in general algebras. In a concurrent paper [4], Hoehnke deals more specifically with radicals in semigroups. This lecture is an account of the simpler aspects of Hoehnke’s theory as it applies to semigroups, but based on Tully’s theory of radicals in semigroups, as set forth in §11.6 of [2]. An address delivered at the Symposium on Topological Semigroups, University of Florida, Gainesville, Florida, April 7–11, 1969.     

Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

On Matrix Rings and Subhereditary Radicals

Abstract

We study some one-sided ideals of row bounded and of column bounded matrix rings over free algebras. Obtained results are applied to answer several open problems on subhereditary radicals of associative rings. In particular we show that the right strongly prime radical is not left subhereditary and that the lattice of left (right) subhereditary radicals is not complete.     

分次根、弱分次根与自反根

本文在一般群分次环范畴中,给出了典型分次根NG,LG,gG,rG和KG的元素刻划.通过建立弱分次根概念,研究了自反根Nref,Lref,gref,rref和Kref的性质,探讨了这些分次根和与它们相对应的自反根的关系．     

Lattices of Radicals of Involution Rings

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We show that certain sublattices are isomorphic to sublattices of the lattice of radicals of rings. We characterize the atoms of certain lattices of radicals of involution rings.1991 Mathematics Subject Classification: 16W10, 16N99, 06B99     

The Classification and Realization of Complete Lie Algebras with Commutative Nilpotent Radicals

in this paper, the classification and realization of complete Lie algebras withcommutative nilpotent radicals are given. From this, some results on the radicals of thesecomplete Lie algebras are obtained.     

广义路代数与点Hopf代数

Most of pointed Hopf algebras of dimension p^m with large coradtical are shown to be generalized path algebras. By the theory of generalized path algebras, the representations, homological dimensions and radicals of these Hopf algebras are obtained. The relations between the radicals of path algebras and connectivity of directed graphs are given.     

TENSOR PRODUCTS OF JACOBSON RADICALS IN NEST ALGEBRAS

This paper studies the tensor product RN RM of Jacobson radicals in nest algebras, and obtains that RN RM = {T∈B(H1 H2) : T(N M)(?)N_ M_, N∈N,M∈M}; and based on the characterization of rank-one operators in RN RM,it is proved that if N, M are non-trivial then RN RM=R if and only if N, M are continuous.