Commuting maps on rank k triangular matrices

• Main Authors: Chooi, Wai Leong, Kwa, Kiam Heong, Tan, Li Yin
• Format: Article
• Published: Taylor & Francis Ltd 2020
• Subjects: QA Mathematics
• Online Access: http://eprints.um.edu.my/36699/

Let n >= 2 be an integer and let F be a field with vertical bar F vertical bar >= 3. Let T-n(F) be the ring of n x n upper triangular matrices over F with centre Z. Fixing an integer 2 <= k <= n,we prove thatan additive map psi: T-n (F) -> T-n(F) satisfies A psi (A) = psi(A)A for all rank k matrices A is an element of T-n(F) if and only if there exist an additive map mu: T-n(F) -> Z, Z is an element of Z and alpha is an element of F in which alpha = 0 when vertical bar F vertical bar > 3 or k < n such that psi(A) = ZA + mu(A) alpha(a(11) + a(nn))E-1n for all A = (a(ij)) is an element of T-n(F). Here, E-1n is an element of T-n(F) is the matrix whose (1, n)th entry is one and zeros elsewhere.