Maps on upper triangular matrices preserving zero products

@article{Słowik2017,
abstract = {Consider $\mathcal \{T\}_n(F)$—the ring of all $n\times n$ upper triangular matrices defined over some field $F$. A map $\phi $ is called a zero product preserver on $\{\mathcal \{T\}\}_n(F)$ in both directions if for all $x,y\in \{\mathcal \{T\}\}_n(F)$ the condition $xy=0$ is satisfied if and only if $\phi (x)\phi (y)=0$. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $\phi $ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $\phi $ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix $x$ into a matrix $x^\{\prime \}$ such that $\lbrace y\in \mathcal \{T\}_n(F)\colon xy=0\rbrace =\lbrace y\in \mathcal \{T\}_n(F)\colon x^\{\prime \}y=0\rbrace $, $\lbrace y\in \mathcal \{T\}_n(F)\colon yx=0\rbrace =\lbrace y\in \mathcal \{T\}_n(F)\colon yx^\{\prime \}=0\rbrace $.},
author = {Słowik, Roksana},
journal = {Czechoslovak Mathematical Journal},
keywords = {zero product preserver; upper triangular matrix},
language = {eng},
number = {4},
pages = {1095-1103},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Maps on upper triangular matrices preserving zero products},
url = {http://eudml.org/doc/294386},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Słowik, Roksana
TI - Maps on upper triangular matrices preserving zero products
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 1095
EP - 1103
AB - Consider $\mathcal {T}_n(F)$—the ring of all $n\times n$ upper triangular matrices defined over some field $F$. A map $\phi $ is called a zero product preserver on ${\mathcal {T}}_n(F)$ in both directions if for all $x,y\in {\mathcal {T}}_n(F)$ the condition $xy=0$ is satisfied if and only if $\phi (x)\phi (y)=0$. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $\phi $ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $\phi $ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix $x$ into a matrix $x^{\prime }$ such that $\lbrace y\in \mathcal {T}_n(F)\colon xy=0\rbrace =\lbrace y\in \mathcal {T}_n(F)\colon x^{\prime }y=0\rbrace $, $\lbrace y\in \mathcal {T}_n(F)\colon yx=0\rbrace =\lbrace y\in \mathcal {T}_n(F)\colon yx^{\prime }=0\rbrace $.
LA - eng
KW - zero product preserver; upper triangular matrix
UR - http://eudml.org/doc/294386
ER -