Some New Results on Pseudo n-Strong Drazin Inverses in Rings

In this paper, we give a further study in-depth of the pseudo $n$-strong Drazin inverses in an associative unital ring $R$. The characterizations of elements $a,b\in R$ for which $aa^{\tiny{\textcircled{\qihao D}}}=bb^{\tiny{\textcircled{\qihao D}}}$ are provided, and some new equivalent conditions on pseudo $n$-strong Drazin inverses are obtained. In particular, we show that an element $a\in R$ is pseudo $n$-strong Drazin invertible if, and only if, $a$ is $p$-Drazin invertible and $a-a^{n+1}\in \sqrt{J(R)}$ if, and only if, there exists $e^2=e\in {\rm comm}^2(a)$ such that $ae\in \sqrt{J(R)}$ and $1-(a+e)^n\in \sqrt{J(R)}$. We also consider pseudo $n$-strong Drazin inverses with involution, and discuss the extended versions of Cline's formula and Jacobson's lemma of this new class of generalized inverses. Likewise, we define and explore the so-called {\it pseudo $\pi$-polar} rings and demonstrate their relationships with periodic rings and strongly $\pi$-regular rings, respectively.