how is it possible for a subring S of a ring R to have a unity, even if R has no unity?
wouldn't the unity in S have to be the unity in R?
Well, you could have a ring R with no unity but contains an element $\displaystyle a\ne0$ such that $\displaystyle a+a=0$ and $\displaystyle a^2=a$; then $\displaystyle S=\{0,a\}$ would be a subring with unity (a would be a unity in S even though it is not a unity in R). Unfortunately I can’t think of a concrete example of such a ring/subring – but I’m guessing it’s possible.