Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 1095-1103, 2017
Maps on upper triangular matrices preserving zero products
Roksana Słowik
Received August 5, 2016. First published October 9, 2017.
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 $x'$ such that $\{y\in\mathcal T_n(F) xy=0\}=\{y\in\mathcal T_n(F) x'y=0\}$, $\{y\in\mathcal T_n(F) yx=0\}=\{y\in\mathcal T_n(F) yx'=0\}$.
Keywords: zero product preserver; upper triangular matrix
