关于交换环上保持伴随矩阵的函数

Functions of Preserving Adjoint Matrices over Commutative Rings

设R是一个交换环,f是R到自身的一个映射。如果f保持R上全矩阵空间(或上三角矩阵空间)中的伴随矩阵,则f称为R上全矩阵空间(或上三角矩阵空间)保持伴随矩阵的函数。探讨了交换环上全矩阵空间和上三角矩阵空间保持伴随矩阵的函数,证明了对于交换环R到自身的任一个映射f,下列条件等价(1)f是R上n阶矩阵空间保持伴随矩阵的函数,(2)f是R上n阶上三角矩阵空间保持伴随矩阵的函数,(3)f=f(1)δ,其中f^(n-1)(1)=f(1)且δ是R的非零自同态。所得的结果拓广了域上的重要结论。

Let R be a commutative ring and f a mapping of R to itself.If f preserves the adjoint matrices in the full matrix space(or the upper triangular matrix space)over R,then f is said to be a function of preserving adjoint matrices in the full matrix space(or the upper triangular matrix space)over R.In this paper,the functions of preserving adjoint matrices in the full matrix space and in the upper triangular matrix space over a commutative ring R are discussed and it is proved that the following are equivalent for any mapping f of R to itself:(1)f is a function of preserving adjoint matrices in the n-order matrix space over R;(2)f is a function of preserving adjoint matrices in the n-order upper triangular matrix space over R;(3)f=f(1)δ,where f^(n-1)(1)=f(1)andδis a nonzero endomorphism of R.The results obtained in this paper expand the important conclusion for field.