Revista de la
Unión Matemática Argentina
New characterization of $(b,c)$-inverses through polarity
Btissam Laghmam and Hassane Zguitti

Volume 69, no. 1 (2026), pp. 109–124    

Published online (final version): December 14, 2025

https://doi.org/10.33044/revuma.4980

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Abstract

Given any ring $R$ with unity $1$ and any $a, b, c \in R$, $a$ is called $(b,c)$-polar if there exist two idempotents $p, q \in R$ such that $p \in bRca$, $q \in abRc$, $pb = b$, $cq = c$, $cap = ca$ and $qab = ab$. These $p$ and $q$ are shown to be unique whenever they exist. The existence of $a^{\|(b,c)}$, the $(b,c)$-inverse of $a$, is shown to be equivalent to $a$ being $(b,c)$-polar, and hence $a^{\|(b,c)}$ is itself unique and expressed in terms of $p$ and $q$. Generalizing results of Koliha–Patrício and Song–Zhu–Mosić, further connections between the $(b,c)$-polar and $(b,c)$-invertible properties are found. Applying these results to bounded linear operators on a Banach space, we also generalize some known results in this setting.

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