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 $1_R$?

A simple example (related to the one by Alex P.) is given by the rings of matrices
$$
\left\{ 
\begin{bmatrix}a&0\\0&0\end{bmatrix} : a \in F
\right\}
= S \subseteq
R = \left\{ 
\begin{bmatrix}a&b\\c&d\end{bmatrix} : a, b, c, d \in F
\right\}
$$
where $F$ might be $\Bbb{Z}, \Bbb{Q}, \Bbb{R}, \dots$.
Here
$$
\begin{bmatrix}1&0\\0&0\end{bmatrix} 
= 1_{S} \ne 1_{R} = 
\begin{bmatrix}1&0\\0&1\end{bmatrix}.
$$
