关于诱导映射的两个保持问题 Two Preserver Problems on Induced Maps

令F是一个域，Mn(F)是F上所有nχn矩阵的集合。如果一个映射f:Mn(F)→Mn(F)被定义如下，f:A=(aij)1→(fij((aij))),∀A∈Mn(F)其中{aij|i,j∈[1,2,...n]}是关于F的函数集，则称f是Mn(F)的由{fij}诱导的映射。如果AB=BA意味着f(A)f(B)=f(B)f(A)，则f被称为保交换矩阵。如果B2=B意味着(f(B))2=f(B)，则f被称为保幂等矩阵。本文我们分别刻画保域上矩阵幂等性及交换性的诱导映射。 Let F be a field, Mn(F) be the set of alln × nmatrices over F. If a map f: Mn(F) → Mn(F) is defined by f:A=(aij)1→(fij((aij))), ∀A∈Mn(F) where {aij|i,j∈[1,2,...n]} are the set of func-tions on F, then f is called a map induced by {fij} on Mn(F). If AB = BA implies f(A)f(B) = f(B)f(A), then f is called preserving commutativity of matrices. If B2=B implies (f(B))2=f(B), then f is called preserving idempotent matrices. In this paper, we characterize induced maps preserving idempo-tence and commutativity of matrices over fields, resprectively.