Idempotence-preserving maps without the linearity and surjectivity assumptions

Let M-n(F) be the space of all n x n matrices over a field F of characteristic not 2, and let P-n(F) be the subset of M-n(F) consisting of all n x n idempotent matrices. We denote by Phi(n)(F) the set of all maps from M-n(F) to itself satisfying A - lambdaB is an element of P-n(F) if and only if phi(A) - lambdaphi(B) is an element of P-n(F) for every A, B is an element of M-n(F) and lambda is an element of F. It was shown that phi is an element of Phi(n) (F) if and only if there exists an invertible matrix P is an element of M-n(F) such that either phi(A) = PAP(-1) for every A is an element of M-n(F), or phi(A) = PA(T)P(-1) for every A is an element of M-n(F). This improved Dolinar's result by omitting the surjectivity assumption and extending the complex field to any field of characteristic not 2. (C) 2004 Elsevier Inc. All rights reserved.