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.