Idempotence-preserving maps without the linearity and surjectivity assumptions (Q1434392)

Let \(M_{n}(F)\) be the space of all \(n\times n\) matrices over a field \(F\) of characteristic not \(2\) and let \(P_{n}(F)\subset M_{n}(F)\) consist of all \(n\times n\) idempotent matrices. The main result of this paper is the following: Theorem 1: Let \(\varphi\:M_{n}(F)\to M_{n}(F)\) satisfying \(A-\lambda B\in P_{n}(F)\) if and only if \(\varphi(A)-\lambda\varphi(B)\in P_{n}(F)\) for every \(A,B\in M_{n}(F)\), \(\lambda \in F\). Then there exists an invertible \(P\in M_{n}(F)\) such that either \(\varphi(A)=PAP^{-1}\) for every \(A\in M_{n}(F)\) or \(\varphi(A)=PA^{\top}P^{-1}\) for every \(A\in M_{n}(F)\). This improves a result obtained by \textit{G. Dolinar} [Linear Algebra Appl. 371, 287--300 (2003; Zbl 1031.15003)] in the case of complex number field under the assumption that \(\varphi\) is surjective.