Maps on upper triangular matrices preserving zero products

Słowik, Roksana

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 1095-1103

Résumé

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 $x'$ such that $\{y\in \mathcal T_n(F)\colon xy=0\}=\{y\in \mathcal T_n(F)\colon x'y=0\}$, $\{y\in \mathcal T_n(F)\colon yx=0\}=\{y\in \mathcal T_n(F)\colon yx'=0\}$.

Détail

DOI : 10.21136/CMJ.2017.0416-16

Classification : 15A99, 16U99
Keywords: zero product preserver; upper triangular matrix