Commuting additive maps and some related maps on triangular matrices / Tan Li Yin

Tan , Li Yin (2022) Commuting additive maps and some related maps on triangular matrices / Tan Li Yin. PhD thesis, Universiti Malaya.

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Abstract

Let F be a ring with identity and let n ⩾ 2 be an integer. Denote by Tn(F) the ring of n_n upper triangular matrices over F with centre Z(Tn(F)) and unity In. Let 1 ⩽ i ⩽ j ⩽ n be integers and let Eij 2 Tn(F) denote the standard matrix unit whose (i, j)th entry is one and zero elsewhere. In this thesis, the following results have been obtained: Let 1 < k ⩽ n be an integer and let F be a field. We characterise commuting additive maps ψ : Tn(F) ! Tn(F) on rank k matrices, i.e., additive maps ψ satisfying ψ(A)A = Aψ(A) for all rank k matrices A 2 Tn(F) and show that • when either k < n or jFj ⩾ 3, there exist λ, α 2 F and an additive map μ : Tn(F) ! F such that ψ(A) = λA + μ(A)In + α(a11 + ann)E1n for all A = (aij) 2 Tn(F), where α 6= 0 only if k = n and jFj = 3, • when k = n ⩾ 4 and jFj = 2, there exist λ, α, β1, β2 2 F, H,K 2 Tn(F) and X1, . . . ,Xn 2 Tn(F) satisfying X1 + _ _ _ + Xn = 0 such that ψ(A) = λA + tr (HtA)In + tr (KtA)E1n + Ψα,β1,β2(A) + Σn i=1 aiiXi for all A = (aij) 2 Tn(F), where tr (A) and At are the trace and the transpose of A respectively, and Ψα,β1,β2 : Tn(F) ! Tn(F) is the additive map defined by Ψα,β1,β2(A) = (αa12 + β1(an−1,n + ann))E1,n−1 + (αan−1,n + β2(a11 + a12))E2n for all A = (aij) 2 Tn(F), iii • when k = n = 3 and jFj = 2, there exist λ, α, β, γ 2 F, H,K 2 T3(F) and X1,X2,X3 2 T3(F) satisfying X1 + X2 + X3 = 0 such that ψ(A) = λA + tr (HtA)I3 + tr (KtA)E13 + Ψα,β(A) + Φγ(A) + Σ3 i=1 aiiXi for all A = (aij) 2 T3(F), where Ψα,β : T3(F) ! T3(F) and Φγ : T3(F) ! T3(F) are the additive maps defined by Ψα,β(A) = α(a23 + a33)E12 + β(a11 + a12)E23, Φγ(A) = γ((a12 + a22)E22 + (a11 + a12 + a23 + a33)E33 + a13(E12 + E23)) for all A = (aij) 2 T3(F), and • when k = n = 2 and jFj = 2, there exist λ1, λ2 2 F and X1,X2 2 T2(F) such that ψ(A) = (a11 + a12)X1 + (a22 + a12)X2 + λ1a12I2 + λ2a12E12 for all A = (aij) 2 T2(F). Let F be a division ring. We classify centralizing additive maps ψ : Tn(F) ! Tn(F) on rank one matrices, i.e., additive maps ψ satisfying ψ(A)A

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