Multiplicative mappings at some points on matrix algebras

Let M(n) be the algebra of all n x n matrices, and let phi : M(n) -> M(n) be a linear mapping. We say that phi is a multiplicative mapping at G if phi(ST) = phi(S)phi(T) for any S, T epsilon M(n) with ST = G. Fix G epsilon M(n), we say that G is an all-multiplicative point if every multiplicative linear bijection phi at G with (I(n)) = I(n) is a multiplicative mapping in M(n), where I(n) is the unit matrix in M(n). We mainly show in this paper the following two results: (1) If G epsilon M(n) with det G = 0, then G is an all-multiplicative point in M(n); (2) If phi is an multiplicative mapping at I(n), then there exists an invertible matrix P epsilon M(n) such that either phi(S) = PSP(-1) for any S epsilon M(n) or phi(T) = PT(tr) P(-1) for any T epsilon M(n). 2010 (C) Elsevier Inc. All rights reserved.