<p>Our aim is to introduce one-sided kinds of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\nabla $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>–Drazin inverse in associative rings <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> with unity, where <Equation ID="Equ2"> <EquationSource Format="TEX">$\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}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">{</mo> <mi>a</mi> <mo>∈</mo> <mi mathvariant="script">R</mi> <mo>:</mo> <mn>1</mn> <mo>-</mo> <mi>a</mi> <mi>u</mi> <mspace width="4pt" /> <mrow> <mi mathvariant="normal">is</mi> <mspace width="4pt" /> <mi mathvariant="normal">a</mi> <mspace width="4pt" /> <mi mathvariant="normal">unit</mi> </mrow> <mspace width="4pt" /> <mrow> <mi mathvariant="normal">for</mi> <mspace width="4pt" /> <mi mathvariant="normal">all</mi> <mspace width="4pt" /> <mi mathvariant="normal">unit</mi> </mrow> <mspace width="4pt" /> <mi>u</mi> <mspace width="4pt" /> <mtext>with</mtext> <mspace width="4pt" /> <mi>u</mi> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mi>u</mi> <mo stretchy="false">}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>is the largest Jacobson radical subring (closed by multiplication by nilpotent elements) of a ring <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>. In particular, left and right <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\nabla $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>–Drazin inverses are defined for elements of a ring. Many characterizations for left (or right) <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\nabla $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>–Drazin invertible elements are established based on idempotents, tripotents, powers and matrix representation forms. Certain expressions for left (or right) <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\nabla $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>–Drazin inverse are given too.</p>

One-sided $\nabla $–Drazin inverses

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2026年3月13日

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M. Tamer Koşan,
Dijana Mosić

摘要

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

One-sided \(\nabla \)–Drazin inverses

摘要