New characterization of $(b,c)$-inverses through polarity

In this paper we introduce the notion of $(b,c)$-polar elements in an associative ring $R$. Necessary and sufficient conditions of an element $a\in R$ to be $(b,c)$-polar are investigated. We show that an element $a\in R$ is $(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the $(b,c)$-polarity is a generalization of the polarity along an element introduced by Song, Zhu and Mosi\'c [14] if $b=c$, and the polarity introduced by Koliha and Patricio [10]. Further characterizations are obtained in the Banach space context.