Traces Of Multiadditive Maps On Rank-s Matrices

Ring theory is an important branch of algebra,and the study of maps on rings helps us to analyze the structure of rings.Let R be a ring with center Z.A map f:R→R is called commuting(resp.,centralizing)on a nonempty subset T of R if f(a)a-af(a)=0(resp.,f(a)a-af(a)∈Z)for each a ∈ T.In 1957,Posner firstly considered commuting and centralizing maps,and proved that a centralizing derivation on a noncommutative prime ring must be a zero map.Posner theorem is considered to be the beginning of the theory of generalized identities.Since then,the theory of generalized identities about the study of commuting and centralizing maps has developed rapidly,and more types of identities have been studied.In 1992,Bresar characterized centralizing additive maps on prime rings which made a breakthrough in the study of commuting and centralizing additive maps.In 2012,inspired by linear preserver problems,Franca focused on the subsets of matrices that are not closed under addition,and characterized commuting additive maps on the set of all n × n invertible matrices(singular matrices,respectively)over a field K.In 2014,Liu characterized the centralizing maps on the set of invertible matrices and the set of singular matrices.Later,many classical identities on prime rings have been extended to the set of invertible matrices,the set of singular matrices,and the set of rank-s matrices.In fact,many classical identities can be seen as the case of zero traces of multiadditive maps.In 2018,Xu and Zhu studied zero traces of multiadditive maps,and proved that invertible matrices can determine the range of traces for multiadditive maps under an assumption.In this paper,we characterize the above conclusions on the set of rank-s matrices.We will prove that rank-s matrices can also determine the range of traces for multiadditive maps under a mild technical assumption.At the same time,we extend several classical identities to the set of rank-s matrices.The paper is divided into three chapters.The first chapter introduces the research background of multiadditive maps and the main results of this paper.The second chapter proves that rank-s matrices can also determine the range of traces for multiadditive maps under a mild technical assumption,and the specific results are as follows.Let m,n,s be integers such that 1<m ≤s<n.Let Mn(D)be the ring of all n × n matrices over a division ring D,M an additive subgroup of Mn(D)and G:Mn(D)m→Mn(D)an m-additive map.We use δ1 to represent the trace function of G,that isδ1(x)=G(x,...,x),x∈Mn(D).If δ1(x)∈M for all x∈Mns(D),then δ1(x)∈M for all x∈Mn(D).On the basis of the main theorem,we obtain several more concise theorems after removing the technical assumption.Meanwhile,we also give an example to show that the main theorem will not be true if s<m.The third chapter is the application of the main conclusions.We also extend Liu theorem,Franca theorem,Beidar-Fong-Lee-Wong theorem and Lee-Wong-Lin-Wang theorem,respectively,to the case of rank-s matrices for m ≤s<n.