# inverses in rings

Let $R$ be a ring with unity $1$ and $r\in R$. Then $r$ is left invertible if there exists $q\in R$ with $qr=1$; $q$ is a left inverse of $r$. Similarly, $r$ is right invertible if there exists $s\in R$ with $rs=1$; $s$ is a 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 are equal, and $r$ is a unit.

 Title inverses in rings Canonical name InversesInRings Date of creation 2013-03-22 17:08:55 Last modified on 2013-03-22 17:08:55 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 4 Author Wkbj79 (1863) Entry type Topic Classification msc 16-00 Related topic Klein4Ring Related topic LeftAndRightUnityOfRing Defines left invertible Defines right invertible Defines left inverse Defines right inverse