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

$$e\cdot a=a\cdot e=a\mathit{\hspace{1em}}\forall a\in R.$$

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 $1\ne 0$” 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