left and right unity of ring

If a ring   (R,+,)left identityPlanetmathPlanetmath element e, i.e. if


then e is called the left unity of R.

If a ring R right identity element e, i.e. if


then e is called the right unity of R.

A ring may have several left or right unities (see e.g. the Klein four-ring).

If a ring R has both a left unity e and a right unity e, then they must coincide, since


This situation means that every right unity equals to e, likewise every left unity.  Then we speak simply of a unity of the ring.

Title left and right unity of ring
Canonical name LeftAndRightUnityOfRing
Date of creation 2013-03-22 15:10:54
Last modified on 2013-03-22 15:10:54
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Definition
Classification msc 20-00
Classification msc 16-00
Related topic InversesInRings
Defines left unity
Defines right unity