Several Hold Maps Of The Matrix Set

Research matrix preserver problems has a wide practical applicable background in every field and its abundant researches having rather strong practical meaning.Suppose that F is a field,n,is an integer and 'n?2.Denote by Mn(F)the set of all n x n matrices over F.If f:Mn(F)?Mn(F))defined by f:B =(bij)(?)(fij(bij))(?)B ? Mn(F),where{fij|i,j?[1,2,...,n]} are the set of functions on F,and[1,n]represented by{1,2 … n},then f is called a map induced by {fij} on Mn(F).For research into preservers problems,people are interested in neither a linear nor additions now.Determine induced maps preserving invariants belongs to this condition.Let f be a map induced by {fij},Denote by K the set of all idempo-tent matrices over F.If B E K)means f(B)? K,then we say f is preserving idempotence.Denote by Sn(F)the set of all n x n,symmetric matrices over F.If A,B,AB?n(IF),means f(AB)? f(A)f(B))then we say is preserving mul-tiplicative.In this paper,we characterize induced maps preserving idempotence matrices over fields,determine induced maps preserving n x n upper triangular idempotant matrices over skew fields,study induced maps preserving multiplicative matrices on sn(F)and at last give results of classical adjoint-commuting injective mappings between upper triangular matrix algebras.