将局部环R上行列式为±1的阵表为1－对合之积

Decomposition of Matrices over Local Ring with Determinants being ±1 into 1-Involutions

文献［１］－［７］研究或综述了将域或体上行列式为±１的ｎ阶矩阵表为１－对合之积的问题。本文给出了局部环Ｒ上行列式为±１的ｎ阶矩阵分解为１－对合之积的因子长度的上界。主要结果是：设ｎ≥２，Ａ∈ＧＬｎ（Ｒ），ｄｅｔＡ＝±１，ｒｅｓＡ＝ｒ。则有（１）若Ａ为非伸缩且ｄｅｔＡ＝（－１）ｒ，那么Ａ至多为ｒ个１－对合之积；（２）若ｄｅｔＡ＝（－１）ｒ＋１，那么Ａ至多为ｒ＋１个１－对合之积；（３）若Ａ为伸缩且ｄｅｔＡ＝（－１）ｒ，那么Ａ至多为ｒ＋２个 １－对合之积。

Abstract In this paper, we consider and get main result as follows: Let n≥2, A∈GLn (R), detA=±1, resA=r. Then(1) if A is not dilatation and detA = (-1)r, then at most r factors are needed; (2)if detA= (-1)r+1, then at most r+1 factors ate needed; (3)if A is dilatation and detA= (-1)r, then at most r+ 2 factors are meeted.