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 .
An element of a ring is the unity iff is an idempotent and regular element.
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|