Common properties of elements in a ring and a Banach algebra and application to the \(\nu \)-convergence

In an associative ring \(\mathcal {R}\) $\mathcal{R}$ with a unit element denoted 1, suppose that a and \(b\in \mathcal {R}\) $b\in \mathcal{R}$ satisfy the following equations: \(\begin{aligned} \left\{ \begin{array}{clc} a^{3} &{}=&{} b a^{2}\\ b^3 &{}=&{} ab^2 \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{a}^3& =\hfill & b{a}^2\\ b^3& =\hfill & a{b}^2\end{array}\right)\end{array}$$ We show that \(1-a\) $1-a$ (resp. a) is (generalized/pseudo) Drazin invertible if and only if \(1-b\) $1-b$ (resp. b) is (generalized/pseudo) Drazin invertible. In the Banach algebra setting, we apply the obtained results to the \(\nu \) $\nu$ -convergence.