交换整环上保持反对称矩阵行列式的函数

On functions of preserving determinants of anti-symmetric matrices over commutative domain

探讨了交换整环上反对称矩阵空间中保持行列式的函数,证明了如下结论:设f是交换整环R到自身的一个映射,n(n≥3)是一个整数.如果n是奇数,那么f是R上n阶反对称矩阵空间的保持行列式的函数当且仅当f是R上的奇函数;如果n是偶数,那么f是R上n阶反对称矩阵空间的保持行列式的函数当且仅当f是R上n阶全矩阵空间的保持行列式的函数当且仅当f=f(1)δ,其中f^(n)(1)=f(1),δ是R上的非零自同态.

The functions of preserving the determinants in the anti-symmetric matrix space over a commutative domain were studied in this paper,and the following result was proved:let f be a mapping of a commutative domain R to itself and n,(n≥3),an integer.If n is odd then f is a function of preserving the determinants in the n-order anti-symmetric matrix space over R if and only if f is an odd function,and if n is even then f is a function of preserving the determinants in the n-order anti-symmetric matrix space over R if and only if f is a function of preserving the determinants in the n-order full matrix space over R if and only if f=f(1)δ,where f^(n)(1)=f(1)andδis a nonzero endomorphism of R.