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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 0 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 identity element”; 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 theorem is apparent.
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