Let \(\varphi\) be an endomorphism of \(K[X]\) (or \(K\langle X\rangle)\) such that, for any nontrivial linear combination \(h\) of elements of \(X\), the image \(\varphi(h)\) is a coordinate polynomial of \(K[X]\) (a primitive element of \(K\langle X\rangle)\). It is true that \(\varphi\) is an automorphism? This problem is answered in the negative by means of a concrete example in the case of an non-algebraically closed field.