Maps on upper triangular matrices preserving zero products

Abstract

Consider T _n (F)—the ring of all n × 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 ∈ T _n (F) the condition xy = 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 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 ∈ T _n (F): xy = 0} = {y ∈ T _n (F): x′y = 0}, {y ∈ T _n (F): yx = 0} = {y ∈ T _n (F): yx′ = 0}.