Let $R$ be a ring and $S$ a proper subring of it. Then there exists five cases in all concerning the possible unities of $R$ and $S$

1. $R$ and $S$ have a common unity.
2. $R$ has a unity but $S$ does not.
3. $R$ and $S$ both have their own non-zero unities but these are distinct.
4. $R$ has no unity but $S$ has a non-zero unity.
5. Neither $R$ nor $S$ have unity.

Examples

1. The ring $\mathbb{Q}$ and its subring $\mathbb{Z}$ have the common unity 1.
2. The subring $S$ of even integers of the ring $\mathbb{Z}$ has no unity.
3. Let $S$ be the subring of all pairs $(a,\,0)$ of the ring $R = \mathbb{Z}\times\mathbb{Z}$ , for which the operations ``$+$ ' and ``$\cdot$ ' are defined componentwise (i.e. $(a,\,b)+(c,\,d) = (a+c,\,b+d)$ , etc.). Then $S$ and $R$ have the unities $(1,\,0)$ and $(1,\,1)$ respectively.
4. Let $S$ be the subring of all pairs $(a,\,0)$ of the ring $R = \{(a,\,2b)|\,\,\,a\in\mathbb{Z}\,\land \,b\in\mathbb{Z}\}$ (operations componentwise). Now $S$ has the unity $(1,\,0)$ but $R$ has no unity.
5. Neither the ring $\{(2a,\,2b)|\,\,\,a,\,b\in\mathbb{Z}\}$ (operations componentwise) nor its subring consisting of the pairs $(2a,\,0)$ have unity.