Centralizing maps on invertible or singular matrices over division rings

On nest modules of matrices over division rings

Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We ...

on nest modules of matrices over division rings

let $ m , n in mathbb{n}$, $d$ be a division ring, and $m_{m times n}(d)$ denote the bimodule of all $m times n$ matrices with entries from $d$. first, we characterize one-sided submodules of $m_{m times n}(d)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $d$. next, we introduce the notion of a nest module of matrices with entries from $d$. we ...

Column-Partitioned Matrices Over Rings Without Invertible Transversal Submatrices

Let the columns of a p× q matrix M over any ring be partitioned into n blocks, M = [M1, . . . , Mn]. If no p × p submatrix of M with columns from distinct blocks Mi is invertible, then there is an invertible p×p matrix Q and a positive integer m ≤ p such that QM = [QM1, . . . , QMn] is in reduced echelon form and in all but at most m − 1 blocks QMi the last m entries of each column are either a...

Triangulaizability of Algebras Over Division Rings

Linear maps preserving or strongly preserving majorization on matrices

For $A,Bin M_{nm},$ we say that $A$ is left matrix majorized (resp. left matrix submajorized) by $B$ and write $Aprec_{ell}B$ (resp. $Aprec_{ell s}B$), if $A=RB$ for some $ntimes n$ row stochastic (resp. row substochastic) matrix $R.$ Moreover, we define the relation $sim_{ell s} $ on $M_{nm}$ as follows: $Asim_{ell s} B$ if $Aprec_{ell s} Bprec_{ell s} A.$ This paper characterizes all linear p...