一类带有非退化不变对称双线性函数的幂零李代数

本文证明了一类带有非退化不变对称双线性函数、幂零指数为Ｎ的李代数 满足：定理１．（ｉ）ｃ（ｇ）＝ｇ~Ｎ; （ｉｉ）ｄｉｍｃ（ｇ）＝ｌ．ｌ是ｇ的生成元个数．定理３给出了这类李代数结构的充分且必要条件．     

泛逻辑的零级泛运算模型的代数性质

讨论泛逻辑的零级泛运算模型的基本代数性质。证明T（x，y，^）是阿基米德型三角范数；泛与运算模型与泛蕴涵运算模型形成一个伴随对；当h∈（0，0．75）时，有界格（[0，1]，∨，∧，，＊，→0，1）做成一个MV-代数；当h∈（0．75，1）时，有界格（[0，1]，∨，∧，＊，→0，1）做成一个乘积代数。进一步，给出了零级泛与运算模型与泛或运算模型的加性生成元与乘性生成元。     

一类新的对称自对偶李代数

本文刻划了一类幂零根基为二步幂零李代数的新的非可解对称自对偶李代数．当其Ｌｅｖｉ因子同构于ｓｌ（２，Ｃ）时，用半单李代数的表示理论具体构造了它们．最后，给出了■２中对称自对偶李代数是ＣＳ李代数的一个判别准则。     

李超代数的泛包络代数同构问题

李超代数的一个性质P叫做关于泛包络代数的不变量,如果对于任意李超代数L,H,只要L具有性质P,并且泛包络代数U(L)和U(H)作为结合超代数是同构的,那么H亦具有性质P.通过讨论李超代数关于泛包络代数的不变量证明了:如果L的幂零长度不超过2,那么L和H是同构的.     

Stone泛补代数

本文定义并研究泛补代数和Stone泛补代数,得到Stone泛补代数的泛补骨架B*(L),素滤子和三元组(B*(L),D*(L),(?)(L))刻画,又得到Stone泛补代数是次直既约Stone泛补代数的次直积的充要条件以及泛补代数是相对Stone泛补代数的特征.     

泛代数的二次扩张

在本文中引入了泛代数的二次扩张的概念，解决了ＴＵ－ＥＣ（Ａ）和ＵＴ－ＥＣ（Ａ）的存在、真类和基数问题，范畴ＴＵ－ＥＣ（Ａ）（及ＵＴ－ＥＣ（Ａ）），并得到了有关二次扩张的几个同构定理，还对一点二次扩张作了讨论。     

关于N（2，2，0）代数的中间幂等元

为了深入研究N（2,2,0）代数的代数结构,在N（2,2,0）代数中建立了中间幂等元的概念,讨论了它的基本性质,给出了中间幂等元关联的集合坞是（S,＊,△,0）的子代数的一个条件．证明了当U（2,2,0）代数中包含一个右零半群时,Mg是幂等元集E（S）的子集．并利用坞定义了一个等价关系．     

关于格的极小生成元集

本文证明了格的极小生成元集一定是最小生成元集且只能是非零完全并既约元全体，证明了分配格具有最小生成元集的必要条件是它满足并无限分配律．本文还证明了完全Ｈｅｙｔｉｎｇ代数具有最小生成元集当且仅当它是强代数格，证明了完备格是强代数格当且仅当它和它的对偶格均是具有最小生成元集的分配格．     

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。     

一类新的对称自对偶李代数

本文刻划了一类幂零根基为二步幂零李代数的新的非可解对称自对偶李代数.当其Levi因子同构于sl(2,C)时,用半单李代数的表示理论具体构造了它们.最后,给出了2中对称自对偶李代数是CS李代数的一个判别准则.     

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．     

The structure of algebraic jordan algebras without nonzero nilpotent elements

An algebraic, linear Jordan algebra without nonzero nil-potent elements is proved to be a subdirect sum of prime Jordan algebras each of which has finite capacity or contains simple subalgebras of arbitrary capacity. If in addition the base field has nonzero character-istic or the algebra satisfies a polynomial identity, then each of the summands is determined to be simple of finite capacity. Further, it is proved that algebraic, PI Jordan algebras without nonzero nilpotent elements are locally finite in the sense that any finitely generated subalgebra has finite capacity.     

Posner and herstein theorems for derivations of 3-prime near-rings

The analog of Posner's theorem on the composition of two derivations in prime rings is proved for 3-prime near-rings. It is shown that if d is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a ? N is such that ax^d = x^da for all x ? N, then a is a central element. As a consequence it is shown that if d\ and d₂ are nonzero derivations of a 2-torsionfree 3-prime near-ring N and x^d1y^d2 = y^d2x^d1 for all x, y ? N, then N is a commutative ring. Thus two theorems of Herstein are generalized     

Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a?J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a?P(a)²J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a?J with generalized inverse b the subtriple generated by {a, b} is strongly associative     

Near-rings with no non-zero nilpotent two-sidedR-subsets

It has been proved that, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and if the annihilator of any non-zero ideal is contained in some maximal annihilator, thenR is a subdirect sum of strictly prime near-rings. Moreover, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and satisfying a.c.c. or d.c.c. on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite subdirect sum of strictly prime near-rings. It is also proved that, ifR is a regular and right duo near-ring that satisfies a.c.c. (or d.c.c.) on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite direct sum of near-ringsR _i (1 i n) where eachR _i is simple and strictly prime.     

Nilpotent sign patterns of a given index

A sign pattern is said to be nilpotent of index k if all real matrices in its qualitative class are nilpotent and their maximum nilpotent index equals k. In this paper, we characterize sign patterns that are nilpotent of a given index k. The maximum number of nonzero entries in such sign patterns of a given order is determined as well as the sign patterns with this maximum number of nonzero entries.     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

Rings radical over subrings

LetR be a ring with a subringA such that a power of every element ofR lies inA. The following results are proved: IfR has no nonzero nil right ideals, neither doesA; if moreoverR is prime,A is also prime. IfR is semiprime Goldie, so isA. IfA has no nonzero nilpotent elements, then the nilpotent elements ofR form an ideal. Finally ifR has no nil right ideals andA is Goldie, thenR is Goldie. This work was supported by the National Research Council of Brazil (CNPq) at the University of Chicago. The author wishes to express his gratitude to Professor I. N. Herstein for his advise and encouragement.     

PI-rings and semiprime rings having faithful modules with krull dimension

It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent. If, moreover, the R-module M and the left ideal_R(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings. Moreover, first, a right Artinian ring R such that rad(R)²=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension. Supported by RFFR grant No. 26-93-011-1544. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 562–572, September–October, 1997.