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


If R consists of the mere 0, then 0 is its unity, since in every ring,  0a=a0=0.  Conversely, if 0 is the unity in some ring R, then  R={0}  (because  a=0a=0aR).

Note.  When considering a ring R it is often mentioned that “…having 10” or that “…with non-zero unity”, sometimes only “…with unity” or “…with ”; all these exclude the case  R={0}.


An element u of a ring R is the unity iff u is an idempotentPlanetmathPlanetmath and regular elementPlanetmathPlanetmath.

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


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