The perturbation of the group inverse under the stable perturbation in a unital ring

Let $\mathbb{R}$ be a ring with unit 1 and $a\in \mathbb{R},\bar{a}=a+\delta a\in \mathbb{R}$ such that a^# exists. In this paper, we mainly investigate the perturbation of the group inverse a^# on $\mathbb{R}$. Under the stable perturbation, we obtain the explicit expressions of \bar{a}^#. The results extend the main results in Xue (2007), and Xue and Chen (2007) and some related results in Xue (2012). As an application, we give the representation of the group inverse of the matrix d&b c&0 on the ring $\mathbb{R}$ for certain $d,b,c\in \mathbb{R}$.