unity (original) (raw)

The unity of a ring (R,+,⋅) 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

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

Note. When considering a ring R it is often mentioned that “…having 1≠0” or that “…with non-zero unity”, sometimes only “…with unity” or “…with ”; all these exclude the case R={0}.

Theorem.

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=u⁢x. Similarly, x=x⁢u. So u is the unity of the ring. The other half of the is apparent.