Idempotence-preserving maps between matrix spaces over fields of characteristic 2

Abstract

Let M(n)(F) be the space of all n x n matrices over a field F of characteristic 2 other than F(2) = {0, 1}, and let P(n)(F) be the subset of M(n)(F) consisting of all n x n idempotent matrices. Let m and n be integers with n >= m and n >= 3. We denote by Phi(n,m)(F) the set of all maps from M(n)(F) to M(m)(F) satisfying that A - lambda B is an element of P(n)(F) implies phi(A) - lambda phi(B) is an element of P(m)(F) for all A, B is an element of M(n)(F) and lambda is an element of F. In this paper, we give a complete characterization of Phi(n,m)(F).