The unity of a ring is the multiplicative identity of the ring, if it has such. The unity is often denoted by , or 1. Thus, the unity satisfies
If consists of the mere 0, then is its unity, since in every ring, . Conversely, if 0 is the unity in some ring , then (because ).
Note. When considering a ring it is often mentioned that “…having ” or that “…with non-zero unity”, sometimes only “…with unity” or “…with ”; all these exclude the case .
Proof. Let be an idempotent and regular element. For any element of we have
and because is no left zero divisor, it may be cancelled from the equation; thus we get . Similarly, . So is the unity of the ring. The other half of the is apparent.
|Date of creation||2013-03-22 14:47:17|
|Last modified on||2013-03-22 14:47:17|
|Last modified by||pahio (2872)|
|Synonym||characterization of unity|