Commuting Mappings On The Hochschild Extension Of An Algebra

The Hochschild extension algebras is an important kind of associative algebras.For example,the trivial extension algebras include the triangular algebras.In this paper,we mainly study the commuting mappings on the Hochschild extension algebras.We get some results as follows:Firstly,we describe the general form of commuting mappings on the Hochschild extension algebras.Then,we characterize the properness of commuting mappiings on a special class of the Hochschild extension algebras with the nontrivial idempotent element p which is mentioned in this paper.In this part,we mainly use the complex computations and the methods of multiple linearization to solve the problem.Secondly,as the application,we give some sufficient conditions under which every commuting mapping on a special Hochschild extension algebra is proper.The theorems gained in this part generalize the corresponding conclusions of the triangular algebras.