中文摘要：

主要是介绍了非交换环的几乎弱稳定秩的概念，并利用它来研究非交换环上的右Hermite环，右Bezout环及初等因子环之间的关系．证明了具有几乎弱稳定秩的满足条件V的右（左）Hermite环是初等因子环；还证明了具有几乎弱稳定秩的满足条件V的右Bezout环是右Hermite环；除此之外还得到了几乎的Exchang环具有几乎弱稳定秩．最后，给出了在具有几乎弱稳定秩且J（R）不为零的右（左）Hermite环上的任意矩阵都可以分解成LUM的乘积，其中L，M为下三角矩阵，U为上三角矩阵．

英文摘要：

We introduce the notion of a ring of almost weakly stable range as a generalization of a ring of weakly stable range. We discussed the relationship between the Hermite ring, Bezout ring and Elementary divisor ring on the noncommutative ring, and proved that a distributive right（left） Hermite ring which has almost weakly stable range and satisfy condition V then R is an elementary divisor ring; R has almost weakly stable range and satisfy condition V R is a right Bezout ring then is a right Hermite ring. We introduce the notion of almost exchange ring and show that an almost exchange ring has almost weakly stable range. Besides, we proved that for any matrix A in a right Hermite ring has almost weakly stable range and with nonzero Jacobson radical J（R） can be decomposed to a LUM, where L, M is a lower triangular matrix, and U is a upper triangular matrix.