有界线性算子的Drazin逆的逆序律

该文讨论了两个有界线性算子乘积的Drazin可逆性及其逆序律,分别在P与PQP可交换(即P²QP=PQP²)和Q与QPQ可交换(即Q²PQ=QPQ²)等条件下,采用空间分解的方法得到了PQ的Drazin可逆性及其逆序律(PQ)^D=Q^DP^D成立的等价条件.     

除环上矩阵乘积广义逆的逆序律

给出了除环上矩阵对的一种等价分解,从而分别导出了A( n) {1 }…A( 1) {1 } (A1) …A( n) ) {1 }及A( n) {1 ,2 }…A( 1) {1 ,2 } (A1) …A( n) ) {1 ,2 }的等价条件.     

Reverse Order Law for the Core Inverse in Rings

In this paper, necessary and sufficient conditions of the one-sided reverse order law \((ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}\), the two-sided reverse order law \((ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}\) and \((ba)^{\tiny {\textcircled {\tiny \#}}}=a^{\tiny {\textcircled {\tiny \#}}}b^{\tiny {\textcircled {\tiny \#}}}\) for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as \((ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}(abb^{\tiny {\textcircled {\tiny \#}}})^{\tiny {\textcircled {\tiny \#}}}\), \(a^{\tiny {\textcircled {\tiny \#}}}=b(ab)^{\#}\) and \((ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}\), are also considered.     

Reverse order law for the Moore-Penrose inverse

In this paper we present new results related to the reverse order law for the Moore-Penrose inverse of operators on Hilbert spaces. Some finite-dimensional results are extended to infinite-dimensional settings.     

非交换主理想整环上三矩阵乘积的Moore-Penrose逆的倒换顺序律

本文研究非交换主理想整环R上三矩阵乘积M-P逆的倒序律成立的刻划问题文中阐述R上矩阵的秩的定义,并利用R上矩阵与 R所嵌入的商除环K上矩阵的秩之间的关系,把文[2]中复矩阵的主要结果直至推广到R上,得到了倒序律成立的若干个刻划.     

加权广义逆矩阵的反序律

研究了在权数矩阵M,N可逆的条件下加权广义逆的反序律,给出了多种表达式.     

关于Drazin逆逆序律的一个注记

本文利用矩阵间秩的关系给出Ｄｒａｚｉｎ逆逆序律成立的一个充分必要条件     

算子乘积的Moore-Penrose逆序律

借助特殊的空间分解,重新刻画算子乘积的Moore-Penrose逆序律成立的充要条件.给出当A,B,AB为闭值域算子时,两个算子乘积Moore-Penrose逆序律成立当且仅当R(A*AB)=R(B)∩R(A*)=R(BB*A*).     

Vaught's Conjecture and Group Rings

We discuss Vaught's conjecture for complete theories of modules over some group rings over the integers and other Dedekind domains, and over pullback rings of Dedekind domains.     

Dedekind Finite Rings and a Theorem of Kaplansky

Abstract

A ring Ris Dedekind Finite(=DF) if xy = 1 implies yx = 1 for all x, yin R. Obviously any subring of a DFring Ris DF. The object of the paper is to generalize, and give a radically new proof of a theorem of Kaplansky on group algebras that are Dedekind finite. We shall prove that all right subrings of right and left self-injective (in fact, continuous) rings are DF.     

任意多个矩阵之积的(T,S,2)-逆的反序律

本文给出了任意r个矩阵之积的(T,S,2)-逆的反序律成立的充要条件.     

关于群中乘积元素的阶

讨论群中两个元素a,b的阶不相等时其乘积ab的阶的一类计算问题.设ㄧaㄧ=m,ㄧ bㄧ=n,若(m,n)=1,且存在k∈N使a=bk,则有ㄧabㄧ=mn/d1d2,其中d1=(m,k+1),d2=(n,k+1).若m≠n,ab=ba,且(m,n)ㄧm/(m,n),或(m,n)ㄧn/(m,n),则有ㄧabㄧ=[m,n].     

The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.     

Further results on the reverse order law for the Moore-Penrose inverse in rings with involution

We present some equivalent conditions of the reverse order law for the Moore-Penrose inverse in rings with involution, extending some well-known results to more general settings. Then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted Moore-Penrose inverse in C^∗-algebras.     

A comment on some recent results concerning the reverse order law for {1, 3, 4}-inverses

This paper has been motivated by the one of Liu and Yang [D. Liu, H. Yang, The reverse order law for {1, 3, 4}-inverse of the product of two matrices, Appl. Math. Comp. 215 (12) (2010) 4293-4303] in which the authors consider separately the cases when (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} and (AB){1,3,4}=B{1,3,4}·A{1,3,4}, where A∈C^n×m and B∈C^m×n. Here we prove that (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} is actually equivalent to (AB){1,3,4}=B{1,3,4}·A{1,3,4}. We show that (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} can only be possible if and in this case, we present purely algebraic necessary and sufficient conditions for this inclusion to hold. Also we give some new characterizations of B{1,3,4}·A{1,3,4}⊆(AB){1,3,4}.     

Decomposition and Group Theoretic Characterization of Pairs of Inverse Relations of the Riordan Type

A new solution to Riordans problem of combinatorial identities classification is presented. An algebgraic characterization of pairs of inverse relations of the Riordan type is given. The use of the integral representation approach for generating new types of combinatorial identities is demonstrated. Supported in part by the National Sciences and Engineering Research Council of Canada on Grant NSERC-108343.Mathematics Subject Classifications (2000) combinatorics, algebra.     

An Example of Bergman's and the Extension Problem for Clean Rings

An example of Bergman is used to show that the extension of a clean ring by another clean ring need not be clean. That is, there exists a ring R and an ideal I of R such that both R/I and I are clean and idempotents lift modulo I, but R is not clean.