Upper Triangular Block Matrix Algebra Idempotent Preserving Linear Operator,

The preserver problem between different sets of matrices has been active, however ,some related references have shown that the results of preserver problem between sets of block triangular matrix are not enough.Suppose R is a commutative principal ideal domain with 1. Let M_n(R)be the module of n full matrices over R, all matrices with formmake up the submodule of M_n(R) , denoted by V ,A_ii expresses u_i square matrix, and n₁+…+n_k = n. When A₁₁=…= A_kk= O , we designate the submodule of V by V₀, and designate the submodule of V by V_i when A_pq are O except A_ii(i = 1,…, k) , thus,obviouslyLetΓ= {P∈V|P² = P},Γ₁ = {P∈M_n(R)|P² = P}. If both Aand B are idempotent matrices inΓand AB = BA = O, then we call that A and B are orthogonal.Let f be an operator from V to M_n(R), if f satisfies f(P) (?)Γ₁ and A and B are orthogonal matrices inΓimplies that f(A)and f(B) are orthogonal matrices inΓ₁,we call f is a preserving idempotent and orthogonal linear operator. We denote the set of f by∑.In this paper,we get the forms of the operator from V to M_n(R) ,from V to V with some conditions.And as application ,we solve the corresponding problems of preserving tripotent and group inverse.