  In the paper, we study the relation between the images of polynomial
derivations and their simplicity. We prove that the images of simple Shamsuddin
derivations are not Mathieu-Zhao spaces. In addition, we also show that the
images of some simple derivations in dimension three are not Mathieu-Zhao
spaces. Thus, we conjecture that the images of simple derivations in dimension
greater than one are not Mathieu-Zhao spaces. We also prove that locally
nilpotent derivations are not simple in dimension greater than one.
