Several Questions About Inducing Retention Mapping

Preserver problems have important theorelical value,and it can be widely used in warious fields.So that the reserach of preserver problems has attracted the attrention of many mathematicians in the past few decades.The study of preserver problems on matrices is more active than other preserver problems.Suppose that F is a field and n>2 is an integer.Denote by Mn(F)the set of all n x n matrices over F.Denote by Tn(F)the set of all n x n upper triangular matrices over F,Denote bySn(F)the set of all n x n symmetric matrices over F.Let ?ij?[1,n])be a function of F,and[1,n]represented by {1,2…n}.If ?defined by?:A =(aij)?(?ij(aij)),(?)A ? Mn(F)(Tn(F)),We say f induced by {fij|ij?[1,n]}.If AA-1 = In,means f(4)f(A-1)? In,then we say f is preserving inverses.If AB = BA,means f(A)f(B)= f(B)f(A),then we say f is preserving commutativ-ity of matrices.In this paper,we respectively characterize induced maps preserving commuta-tivity on n x n matrices and upper triangular matrices over fields,then we also respectively characterize induced maps preserving inverses on symmetric matrices and upper triangular matrices over fields.