Mathieu Subspaces Theory Of Polynomial Algebras And Matrix Algebras

The Mathieu subspaces is a natural generalization of ideals,arising from the study of the Jacobi conjecture.The definition was gived by Wenhua Zhao in 2010,and the theory of Mathieu subspaces is at an initial stage at present,In this thesis,the first chapter introduces the definition,background and development of Mathieu subspaces;In the second chapter we describe the relation among Mathieu subspace,Jacobi conjecture and the images of differential operators(including derivations);The third chapter gives a detailed description of the vanishing conjecture,the image conjecture of the differential operator,and the results concerning the images of derivations.In the fourth chapter,we introduce the general theory of Mathieu subspace;And in the last chapter,we introduce the Mathieu subspaces theory of one-variable polynomial algebras and matrix algebras.