inverses in ringsInversesInRings2007-05-24 11:11:042007-05-24 11:11:04Topicringleft invertibleright invertibleleft inverseright inverse\usepackage{amssymb}
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Let $R$ be a ring with unity $1$ and $r \in R$. Then $r$ is \emph{left invertible} if there exists $q \in R$ with $qr=1$; $q$ is a \emph{left inverse} of $r$. Similarly, $r$ is \emph{right invertible} if there exists $s \in R$ with $rs=1$; $s$ is a \emph{right inverse} of $r$.
Note that, if $r$ is left invertible, it may not have a unique left inverse, and similarly for right invertible elements. On the other hand, if $r$ is left invertible and right invertible, then it has exactly one left inverse and one right inverse. Moreover, these two \PMlinkescapetext{inverses} are equal, and $r$ is a unit.