一类交换环上全矩阵半群的自同构

Automorphisms of the Full Matrix Semigroup over a Class of Commutative Rings

设R是有1的连通交换环,Mn(R)是R上所有n×n矩阵组成的矩阵乘法半群,Φ是Mn(R)上的任一半群自同构.证明了若R上的幂等矩阵均可相似对角化,则存在可逆矩阵P∈Mn(R),环R的自同构θ,使得Φ(A)=PAθP-1,A∈Mn(R).

Let Rbe a connected commutative ring with 1,Mn（R）the multiplicative semigroup consisting of all n×n matrices over R,Φan arbitrary automorphism of Mn（R）.In this paper,we prove that if every idempotent matrix over Ris similar to a diagonal matrix,then there exist an invertible matrix P∈Mn（R）and an automorphismθ of R,such thatΦ（A）=PAθP^-1,A∈Mn（R）.