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="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>∇</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> <mml:mo>:</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>a</mml:mi> <mml:mi>u</mml:mi> <mml:mspace/> <mml:mrow> <mml:mi>is</mml:mi> <mml:mspace/> <mml:mi>a</mml:mi> <mml:mspace/> <mml:mi>unit</mml:mi> </mml:mrow> <mml:mspace/> <mml:mrow> <mml:mi>for</mml:mi> <mml:mspace/> <mml:mi>all</mml:mi> <mml:mspace/> <mml:mi>unit</mml:mi> </mml:mrow> <mml:mspace/> <mml:mi>u</mml:mi> <mml:mspace/> <mml:mtext>with</mml:mtext> <mml:mspace/> <mml:mi>u</mml:mi> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mi>u</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> is the largest Jacobson radical subring (closed by multiplication by nilpotent elements) of a ring $$\mathcal {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> . In particular, left and right $$\nabla $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> –Drazin inverses are defined for elements of a ring. Many characterizations for left (or right) $$\nabla $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> –Drazin invertible elements are established based on idempotents, tripotents, powers and matrix representation forms. Certain expressions for left (or right) $$\nabla $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∇</mml:mi> </mml:math> –Drazin inverse are given too.