unity of subring

Theorem.

Let $S$ be a proper subring of the ring $R$.  If $S$ has a non-zero unity $u$ which is not unity of $R$, then $u$ is a zero divisor of $R$.

Proof.  Because $u$ is not unity of $R$, there exists an element $r$ of $R$ such that  $ru\ne r$.  Then we have  $(ru)u=r(uu)=ru$, which implies that  $0=(ru)u-ru=(ru-r)\cdot u$.  Since neither  $ru-r$  nor  $u$  is 0, the element  $u$  is a zero divisor in $R$.