# unity

The unity of a ring  $(R,\,+,\,\cdot)$  is the multiplicative identity of the ring, if it has such.  The unity is often denoted by $e$, $u$ or 1.  Thus, the unity satisfies

 $e\cdot a\;=\;a\cdot e\;=\;a\quad\forall a\in R.$

If $R$ consists of the mere 0, then $0$ is its unity, since in every ring,  $0\cdot a=a\cdot 0=0$.  Conversely, if 0 is the unity in some ring $R$, then  $R=\{0\}$  (because  $a=0\cdot a=0\,\,\forall a\in R$).

Note.  When considering a ring $R$ it is often mentioned that “…having $1\neq 0$” or that “…with non-zero unity”, sometimes only “…with unity” or “…with ”; all these exclude the case  $R=\{0\}$.

###### Theorem.

An element $u$ of a ring $R$ is the unity iff $u$ is an idempotent and regular element.

Proof.  Let $u$ be an idempotent and regular element.  For any element $x$ of $R$ we have

 $ux\;=\;u^{2}x\;=\;u(ux),$

and because $u$ is no left zero divisor, it may be cancelled from the equation; thus we get  $x=ux$.  Similarly,  $x=xu$.  So $u$ is the unity of the ring.  The other half of the is apparent.

 Title unity Canonical name Unity Date of creation 2013-03-22 14:47:17 Last modified on 2013-03-22 14:47:17 Owner pahio (2872) Last modified by pahio (2872) Numerical id 15 Author pahio (2872) Entry type Definition Classification msc 20-00 Classification msc 16-00 Classification msc 13-00 Synonym multiplicative identity Synonym characterization of unity Related topic ZeroDivisor Related topic RootOfUnity Related topic ZeroRing Related topic NonZeroDivisorsOfFiniteRing Related topic OppositePolynomial Defines non-zero unity Defines nonzero unity