Invertible linear maps preserving {1}-inverses of matrices over PID

Let R be a PID,chR = 2,n > 1, M n(R) be then xn full matrix algebra over R.f denotes any invertible linear map preserving {1}-inverses from M n(R) to itself. In this paper, we have proven thatf is an invertible linear map on M n(R) preserving {1}-inverses if and only iff satisfies any one of the following two conditions: (i) there exists a matrixP ε GL n(R) such that f(A) =PAP -1 for allA ε M n(R), (ii) there exists a matrixP ε GL n(R) such thatf(A) =PA t P -1 forA ε M n(R). © 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.