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# 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 inverses are equal, and $r$ is a unit.

Defines:

left invertible, right invertible, left inverse, right inverse

Related:

Klein4Ring, LeftAndRightUnityOfRing

Major Section:

Reference

Type of Math Object:

Topic

Parent:

## Mathematics Subject Classification

16-00*no label found*

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## Comments

## left or right inverse

If an element j has a unique left or right inverse, h it is the inverse for the element.