Adjacency Preserving Additive Bijections On The Block Triangular Matrices Over A Bezout Domain

The preserving problems is an active topic in the matrix algebra, and its mainproblem is to characterize the additive operators which preserves invariant on matrixspaces. The geometry of matrices was initiated by Hua L. K. in the mid forties oflast century. In recent years, the geometry of matrices have a new development, andadditive preserving problems have a close relationship with the geometry of matrices.The geometry of matrices can be regarded as a study of deeper level, and the additivepreserving problems can be regarded as the preparatory (preliminary) work of thegeometry of matrices.Present there is not the fundamental theorem of the geometry of the block triangu-lar matrices over a Bezout domain, thus this paper researches the adjacency preservingadditive bijections on the block triangular matrices over a Bezout domain. Firstly, bythe method of the geometry of matrices, we study the arithmetic distance, the distance,and the geometric structure and basic properties of maximal sets. Secondly, this papergets some basic properties about the adjacency preserving additive bijections on theblock triangular matrices over a Bezout domain. Lastly, this paper proves the followmain results:Let R be a Bezout domain, n₁,n₂ be integers≥2, and T_n(i1)2 be the set of the 4×4block triangular matrices over R. Let (?) : T_n(i1)2→T_n(i1)2 be an adjacency preservingadditive bijection in both directions, i.e. rank(X -Y ) = 1 (?) rank((?)(X)-(?)(Y )) = 1(?)X,Y∈T_n(i1)2. If T_n(i1)2 = T(n(3-i₁)2), then (?) is of the formwhere P,Q∈T_n(i1)2(R) are fixed invertible matrices, andσis an automorphism of R.If T_n(i1)2 = T(n(3-i₁)2), then (?) is of the form either above or(?)(X) = P(X+)Ï„Q, (?)X∈T_n(i1)2,where P,Q∈T_n(i1)2 are fixed invertible matrices, andÏ„is an anti-automorphism of R.