inverses in rings

Let R be a ring with unity 1 and rR. Then r is left invertible if there exists qR with qr=1; q is a left inverseMathworldPlanetmath of r. Similarly, r is right invertible if there exists sR 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