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

摘要

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. ε~2 = ε, S ∈ T is invertible, P = Σ_(i=1) ~n E_(i,n-i+1), 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.