Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R.
Let $a \in S$. Then $1_S*a = a$ and this a exists in R so that $1_R*a=a$
$1_S*a - 1_R*a = 0$
2) a = 0
3) $1_S-1_R \neq 0$ and $a \neq 0$
Then I'm not sure what to do. I don't even know if this is right. Can somebody please help?