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Maps on upper triangular matrices preserving zero products

Roksana Słowik — 2017

Czechoslovak Mathematical Journal

Consider $𝒯_{n} (F)$ —the ring of all $n \times n$ upper triangular matrices defined over some field $F$ . A map $φ$ is called a zero product preserver on $𝒯_{n} (F)$ in both directions if for all $x, y \in 𝒯_{n} (F)$ the condition $x y = 0$ is satisfied if and only if $φ (x) φ (y) = 0$ . 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 $φ$ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $φ$ as a composition...

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