The N-Commuting Maps On Incidence Algebras

Let R be a commutative ring with identity element,and A be an associative R-algebra with center Z(A).For any x,y?A and any positive integer n,we set[y,x]₁:=[y,x]=yx-xy and recursively[y,x]_n:=[[y,x]_n-1,x].An R-linear map?:A?A is said to be n-commuting on A if[?(x),x]_n=0.The purpose of this dissertation is to study n-commuting maps on incidence algebras,and the main topic belongs to a branch of the field of rings and algebras named functional identities,which to some extent is a generalization of polynomial identities.In the theory of functional identities,n-commuting maps are important for studying the associated Lie structure defined by commutators of an associative algebra,especially for characterizing the Lie type isomorphisms.Let I(X,R)be the incidence algebra defined on a locally finite pre-ordered set X over R,and?be an n-commuting map on I(X,R).In this dissertation,depending on the techniques of linear algebras and combinatorics,we obtain the main result as follows:if R is 2-torsion and n!-torsion free,and if any two edges(forgetting directions)in each connected components of the complete Hasse diagram associated to X are contained in a certain cycle,then?is proper,i.e.,?(f)=?f+?(f),(?)f?I(X,R),where??Z(I(X,R)),and?:I(X,R)?Z(I(X,R))is an R-linear map.It would to helpful to state that when X is a finite pre-ordered set,the above conclusion still holds provided that R is 2-torsion free.This dissertation is organized as follows.The first chapter is the introduction,which mainly introduces the relevant background and the framework of this thesis.In the second chapter,we introduce incidence algebras and some related combinatorics of graphs.The third chapter is the central part,in which we study n-commuting maps on incidence algebras in details.In the fourth chapter some related problems needing further studies are listed.