2. Well, you could have a ring R with no unity but contains an element $a\ne0$ such that $a+a=0$ and $a^2=a$; then $S=\{0,a\}$ 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.