Modular Automorphisms of Triangular Matrices over Commutative Rings Preserving Involutory and Tripotent

摘要

Suppose R is a commutative ring with 1, and 2 is a unit of R. LetTn(R) be the n× n upper triangular matrix modular over R, and let Li(R) (i=2 or 3) be the set of all R-module automorphisms on Tn(R) that preserve involutory or tripotent. The main result in this paper is that f∈Li(R) if and only if there exists an invertible matrix U∈Tn(R) and orthogonal idempotent elements el, e2, e3 and e4 in R with 4↑∑i=1 = 1 such that f(X) =U ((e1 -e2)x + (e3 -e4)X^δ) U^-1' VX∈Tn(R),where X^δ = (Xn+1-j n+1-i).