# left and right unity of ring

If a ring   $(R,\,+,\,\cdot)$left identity element $e$, i.e. if

 $e\cdot a=a\quad\forall a,$

then $e$ is called the left unity of $R$.

If a ring $R$ right identity element $e^{\prime}$, i.e. if

 $a\cdot e^{\prime}=a\quad\forall a,$

then $e^{\prime}$ 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 $e^{\prime}$, then they must coincide, since

 $e^{\prime}=e\cdot e^{\prime}=e.$

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 LeftAndRightUnityOfRing 2013-03-22 15:10:54 2013-03-22 15:10:54 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Definition msc 20-00 msc 16-00 InversesInRings left unity right unity