unities of ring and subring

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.

Note:   In the cases 3 and 4, the unity of the subring $S$ must be a zero divisor of $R$.

Examples

1. The ring $\mathbb{Q}$ and its subring $\mathbb{Z}$ have the commonunity 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 $(a,\mathrm{\hspace{0.17em}0})$ 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 $(1,\mathrm{\hspace{0.17em}0})$ and $(1,\mathrm{\hspace{0.17em}1})$, respectively.

4. Let $S$ be the subring of all pairs $(a,\mathrm{\hspace{0.17em}0})$ of the ring  $R=\{(a,\mathrm{\hspace{0.17em}2}b)|a\in \mathbb{Z}\wedge b\in \mathbb{Z}\}$ (operations componentwise).  Now $S$ has the unity $(1,\mathrm{\hspace{0.17em}0})$ but $R$ has no unity.

5. Neither the ring $\{(2a,\mathrm{\hspace{0.17em}2}b)|a,b\in \mathbb{Z}\}$ (operations componentwise) nor its subring consisting of the pairs $(2a,\mathrm{\hspace{0.17em}0})$ have unity.