left and right unity of ring

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

$$e\cdot a=a\mathit{\hspace{1em}}\forall a,$$

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

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

$$a\cdot e^{\prime }=a\mathit{\hspace{1em}}\forall a,$$

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.