正则环上矩阵分解

利用幂等矩阵和单边可逆阵,给出了正则环上具有群逆的矩阵结构,并证明了具有奇数特征的单边单位正则环上的矩阵都可分解为两个单边可逆矩阵和形式。     

加法范畴中态射的Drazin逆

本文研究了加法范畴上态射的Drazin逆。首先给出了态射和φ η与态射φ有Drazin逆的一个关系，得到了φ η的Drazin逆的一个公式，其次证明了态射φ有Drazin逆当且仅当φ^k有群逆（k为某一正整数）。最后还证明了：如果2为可逆态射，则具有Drazin逆的态射一定为两个可逆态射之和。     

关于四个三幂等阵线性组合的可逆性和群逆

本文研究了四个三幂等阵线性组合的可逆性及群逆.利用矩阵分解的方法,获得了它们可逆及群逆的一些条件,并得到其逆和群逆的计算公式,这些结论完善了k幂等阵可逆性理论.     

交换环上矩阵的Drazin逆

本文应用子式讨论交换环上矩阵的Drazin逆和群逆,给出了矩阵A的Drazin逆和群逆的整体和单个元素的表达式.     

群可逆元的一些刻画

研究了群可逆元的一些性质,给出群可逆元的一些刻画,进一步研究正则元是群可逆元的若干条件.     

具有对合的自反环（英文）

本文研究了具有对合的环的自反性质.称环R的一个对合*是自反的,如果对任意a,b∈R,由aRb=0可推出bRa~*=0.若环R具有自反的对合*,则称R为*-自反环.我们对*-自反环的性质进行了刻画,并给出了一些具体的例子.作为应用,我们主要研究了与*-自反环相关的广义逆.对*-自反环R,我们证明了Moore-Penrose可逆元未必是群可逆元.     

群逆和Drazin逆的一种新的表达式及其应用

本文研究了群逆的存在条件及群逆、Drazin逆的表示与计算.利用行列式表示方法,得到了群逆存在的充要条件,给出了群逆的与原矩阵最大非奇异子阵有关的表达式.并推广到Drazin逆.为群逆和Drazin逆的计算提供了一类新的算法.     

环上矩阵广义Moore-Penrose逆的存在性

讨论带有对合反自同构*有单位元的结合环R上矩阵的广义Moore-Penrose 逆,给出了环R上矩阵的广义Moore-Penrose逆存在的几个充要条件．特别,得到了环 R上矩阵A的关于M和N的广义Moore-Penrose逆存在的充要条件是A有分解A= GDH,其中D2=D,(MD)*=MD,(GD)*MGD+M(I-D)和DHN-1(DH)*+ (I-D)M-1均可逆．     

无单位元的群分次环,Smaxh积及其一个应用

本文对于具有局部单位元的群分次环Ｒ证明了在Ｒ＃Ｇ模范畴与分次左Ｒ－模范畴是同构的，并给出Ｒ是分次右完全环的一些充要条件。     

关于环上矩阵的群逆与Drazin逆

本文给出了环上一类方阵有群逆，｛１，５｝－道的充要条件及其它们的表式，推广了体（域）上关于群逆的Ｃｌｉｎｅ定理．此外还首次得到了矩阵有Ｄｒａｚｉｎ逆的判别准则及其它的表式．     

Elements in one-sided unit regular rings

Let R be regular. We show that the following are equivalent:(1) R is a one sided unit regular ring. (2) For every x [euro] R, there exist an idempotente and a right or left invertible u such that x [d] eu or x [d] ue. (3) For every x [euro] R,there exists a right or left invertible u such that xu or ux is an idempotent. Moreover, we give some characterizations of one-sided unit regular rings by group inverses.     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.     

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.     

Properties of m-EP operators

In this paper, we study the characteristics of m-EP operators and the properties of the particular case of m-EP operators that are Drazin invertible. The relations between the Moore–Penrose inverses and Drazin inverses are built by m-EP and many closely equivalent relations are investigated by using appropriate idempotents.     

Pseudo core inverses in rings with involution

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary ?-ring case by Raki?, Din?i? and Djordjevi? in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of ?-ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field Fq2, which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for an m×n matrix A over Fq2 having an M-P inverse are obtained, which make clear the set of m×n matrices over Fq2 having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Weighted core–EP inverse of an operator between Hilbert spaces

We introduce and study the weighted core–EP inverse of an operator between two Hilbert spaces as a generalization of the weighted core–EP inverse for a rectangular matrix. Several new properties of weighted core–EP inverses are given and some known results are extended. Using a weighted operator, the core–EP pre-order and the minus partial order of corresponding operators, we define new pre-orders on the set of all Wg–Drazin invertible operators between two Hilbert spaces. As consequences of our results, we present a new characterization and new representations of the core–EP inverse, new characterizations of the core–EP pre-order and extend the core–EP pre-order to a partial order.     

The group inverse and core inverse of sums of two elements in a ring

Abstract

In Dedekind-finite ring, we present the group inverse of sum of two group invertible elements under different conditions. Then, the core inverse of a sum of two core invertible elements is investigated. Furthermore, the core inverse of the difference of two core invertible elements is presented. These results generalized the corresponding results of complex matrices.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.