Identity of Rings How would I show that there is a ring $R$ with identity $1_R$ and a subring $S$ not containing $1_R$, but such that $S$ has its own identity $1_S$ not equal to $1_R$?

You can take $R = \mathbb{Z} \times \mathbb{Z}$ and $S = \mathbb{Z} \times \{0\} \subset R$. Then $1_R = (1,1) \neq (1,0) = 1_S$.
