Ring with unity 1 that has subring with unity 1'

Problem: Give an example of a ring with unity 1 that has a subring with unity $1' \not= 1$.
let $R$ be the ring of all $2 \times 2$ matrices with integer entries. Let $S=\left \{ \begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}: \ a \in \mathbb{Z} \right \}.$ then $S$ is a subring of $R$ and $1_S = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \neq 1_R.$