Geometry of block triangular matrices over a division ring

Abstract

Let D be a division ring with D ≠ F 2 and T ( n i , k ) denote the set of k × k block triangular matrices over D. Let ϕ be a bijective map from T ( n i , k ) ( k , n 1 , n k ⩾ 2 ) to itself such that both ϕ and ϕ −1 preserve the adjacency. By the method of maximal sets of rank one and affine geometry, we characterize ϕ and obtain the fundamental theorem of the geometry on T ( n i , k ) . As a corollary, weakly block-additive adjacency preserving bijective maps in both directions on T ( n i , k ) are characterized. As applications of the fundamental theorem, ring automorphisms or ring anti-automorphisms of T ( n i , k ) are characterized, and Jordan automorphisms of Jordan ring J ( T ( n i , k ) ) are also characterized.