One-sided $$\nabla $$–Drazin inverses

Abstract Our aim is to introduce one-sided kinds of the $$\nabla $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> –Drazin inverse in associative rings $$\mathcal {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> with unity, where $$\begin{aligned} \nabla (\mathcal {R})=\{a\in \mathcal {R}: 1-au \ \mathrm{is \ a \ unit} \ \mathrm{for\ all \ unit }\ u\ \textrm{with}\ ua=au\} \end{aligned}$$ <mml:math xmlns:mml=