The left $(b,c)$-core inverse in rings and its applications

Abstract

Let $R$ be a $*$-ring and let $a,b,c\in R$. The paper aims to introduce and investigate the left $(b,c)$-core inverses of $a$. The element $a\in R$ is left $(b,c)$-core invertible if there exists some $x\in R$ such that $caxc=c$, $xcab=b$ and $cax=(cax)^*$. Such an $x$ is called a left $(b,c)$-core inverse of $a$. Several criteria for the left $(b,c)$-core inverse are given. Among of these, it is proved that $a$ is left $(b,c)$-core invertible if and only if $a$ is left $(b,c)$-invertible and $c$ (or $ca$) is $\{1,3\}$-invertible, under certain condition. Moreover, the connection between left $(b,c)$-core inverses and left $(b,c)$-inverses is established. Finally, $y=xcax$ is the $(b,c)$-core inverse of $a$ if and only if the descending chain $caR\supseteq (ca)^{2}yR \supseteq \cdots \supseteq (ca)^{n+1}y^{n}R\supseteq \cdots$ stabilizes. Finally, the application of this type of generalized inverses is given to carbon emission.