If a ring $(R,\,+,\,\cdot)$ , contains a multiplicative 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 $R$ contains a multiplicative right identity element $e'$ i.e. if $$a\cdot e' = a \quad \forall a,$$ 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 $$e' = e\cdot e' = e.$$ This situation means that every right unity equals to $e$ likewise every left unity. Then we speak simply of a unity of the ring.