Hello everyone. I was wondering if anyone could think of an example of a subring of a commutative ring (with unity), that does not have the same unity?

I can think of one with a non-commutative ring:

$\displaystyle \left\{ \left( \begin{array}{cc} x & 0 \\ 0 & 0 \\ \end{array} \right) } \mid x \in \mathbb{R} \right\}$ which has a different unity element from $\displaystyle M_2(\mathbb{R})$.

Thanks for any help