交换整环上矩阵模之间保幂等的线性映射及其应用

Linear maps preserving idempotence between matrix modules over commutative integral domains and its applications

设R是一个含有单位元1的交换整环,Mn(R)是R上的n×n矩阵模,用Pn(R)记Mn(R)中所有幂等阵构成的集合。若线性映射f:Mn(R)→Mm(R)满足f(Pn(R))■Pm(R),则称f是保幂等的线性映射。用χR(n,m)表示所有这样保幂等的线性映射的集合。用生成元的定义关系来刻画当n,m≥2,R≠F时χR(n,m)中算子的结构,作为它的应用,得到保矩阵逆的算子结构。

Let R be a commutative integral domains with 1, and Mo （R） be the tt x n matrix modules over R. Let Pn （R） be the subset ofMn（R） consisting of all idempotent matrices. A linear mapf：Mn（R）→Mn（R） is said to preserve idempotence iff（P,（R）） CPm（R）. Denote by Rn（n,m） the set of all such linear maps. The structure of R （n,m） is characterized by defining relations of generators when n, m≥2 and R ≠ F2. As its applications, the structure of operators preserving inverses of matrices are described.