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