The group of commutativity preserving maps on upper triangular matrices over a commutative ring

Abstract

Let 𝒯 = 𝒯 n (R) be the associative algebra of all n × n upper triangular matrices over a unital commutative ring R with n > 2. A map σ on 𝒯 is called preserving commutativity in both directions if xy = yx ⇔ σ(x)σ(y) = σ(y)σ(x). For an invertible linear map σ on 𝒯, the following two conditions are shown to be equivalent: (a) σ preserves commutativity in both directions and (b) σ takes the form: σ(X) = cS −1[ϵX + (ϵ − 1)PX′P]S + f(X)I, ∀X ∈ 𝒯, where c ∈ R is invertible, ϵ ∈ R is idempotent, i.e. ϵ2 = ϵ, S ∈ 𝒯 is invertible, , X′ means the transpose of X and f is a linear function from 𝒯 to R such that 1 + f(I) is invertible. This result extends the main theorem of Marcoux and Sourour [L.W. Marcoux and A.R. Sourour, Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras, Linear Algebra Appl. 288 (1999), pp. 89–104] to an arbitrary commutative ring.