On stable range one matrices

On rank range of interval matrices

An interval matrix is a matrix whose entries are intervals in the set of real numbers. Let $p , q $ be nonzero natural numbers and let $μ=( [m_{i,j}, M_{i,j}])_{i,j}$ be a $p \times q$ interval matrix; given a $p \times q$ matrix $A$ with entries in the set of real numbers, we say that $ A \in μ$ if $a_{i,j} \in [m_{i,j}, M_{i,j}] $ for any $i,j$. We establish a criterion to say if an interval matrix contains a matrix of rank $1$. Moreover we determine the maximum rank of the matrices contained in a given interval matrix. Finally, for any interval matrix $μ$ with no more than $3$ columns, we d…

秩在某一范围内的矩阵仿射子空间

Affine subspaces of matrices with rank in a range

本文研究域上秩介于两数间的矩阵仿射子空间最大维数，涵盖恒定秩、秩有上下界的情况，推广实矩阵结果并得到反对称矩阵与行阶梯形矩阵相关结论。

稳定等价与同调维数

Stable equivalences and homological dimensions

本文研究了中心化矩阵代数之间的稳定等价，并证明了相关同调维数的保持以及Alperin--Auslander/Auslander--Reiten猜想的成立。

并非所有 GKK $τ$-矩阵都是稳定的

Not all GKK $τ$-matrices are stable

本文构造了一类不稳定的 GKK $τ$-矩阵，从而否定了关于 GKK 矩阵和 $τ$-矩阵正稳定性的四个猜想。

一个被非负秩一矩阵扰动的奇异M矩阵具有正的主 minors；它是否D稳定？

A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?

研究奇异M矩阵在被非负秩一扰动后的正稳定性与D稳定性，特别情况下可以证明正稳定性或D稳定性，但一般情况无法成立，通过反例说明。然而，此类矩阵被证明为P矩阵。

An infinite analogue of rings with stable rank one

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings), and include examples like the ring of endomorphisms of a vector space over a field F, and the ring of all row- and column- finite matrices over F. We show that the category of QB-rings is stable under the formation of corners, ideals and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings t…

A Bezout ring of stable range 2 which has square stable range 1

In this paper we introduced the concept of a ring of stable range 2 which has square stable range 1. We proved that a Hermitian ring $R$ which has (right) square stable range 1 is an elementary divisor ring if and only if $R$ is a duo ring of neat range 1. And we proved that a commutative Hermitian ring $R$ is a Toeplitz ring if and only if $R$ is a ring of (right) square range 1.

环上矩阵的算术

Arithmetic of matrices over rings

本书研究交换有限生成主理想整环上矩阵环的算术，构建矩阵分解理论，并阐明其与完全线性群子群性质及单侧等价标准形的关系。