Abstract:The algebraic theory of derivations plays an important role in the research of some problems in affine algebraic geometry, for example the Hilbert’s 14th problem, cancellation problems for affine varieties, and the Jacobian conjecture. In this talk, we focus on the problem of whether the images of certain differential operators and derivations are the so-called Mathieu-Zhao subspaces (a generalization of ideals); this problem arose from the Jacobian conjecture. We will introduce the known results and some of our recent work on this problem.