One-sided $$ abla $$–Drazin inverses
M. Tamer Kosan, Dijana Mosić · Ricerche di Matematica · 2026
Abstract Our aim is to introduce one-sided kinds of the $$ abla $$ ∇ –Drazin inverse in associative rings $$\mathcal {R}$$ R with unity, where $$\begin{aligned} abla (\mathcal {R})=\{a\in \mathcal {R}: 1-au \ \mathrm{is \ a \ unit} \ \mathrm{for\ all \ unit }\ u\ \textrm{with}\ ua=au\} \end{aligned}$$ ∇ ( R ) = { a ∈ R : 1 - a u is a unit for all unit u with u a = a u } is the largest Jacobson radical subring (closed by multiplication by nilpotent elements) of a ring $$\mathcal {R}$$ R . In particular, left and right $$ abla $$ ∇ –Drazin inverses are defined for elements of a ring. Many characterizations for left (or right) $$ abla $$ ∇ –Drazin invertible elements are established based on idempotents, tripotents, powers and matrix representation forms. Certain expressions for left (or right) $$ abla $$ ∇ –Drazin inverse are given too.