Journal of the
Korean Mathematical Society JKMS

A ring $R$ is a weakly stable ring provided that $aR+bR=R$ implies that there exists $y\in R$ such that $a+by\in R$ is right or left invertible. In this article, we characterize weakly stable rings by virtue of $2\times 2$ invertible matrices over them. It is shown that a ring $R$ is a weakly stable ring if and only if for any $A\in GL_2(R)$, there exist two invertible lower triangular $L$ and $K$ and an invertible upper triangular $U$ such that $A=LUK$, where two of $L,U$ and $K$ have diagonal entries $1$. Related results are also given. These extend the work of Nagarajan et al.

Keywords: weakly stable ring, invertible matrix, factorization

MSC numbers: 15A23, 16E50