Specific subring with non-zero unity element

Question

I'm trying to find an example of a ring with unity $1\ne 0$ that has a subring with non-zero unity $1' \ne 1$.

Any hints here?

Thanks a lot, Mariogs

Answer 1

Choose any ring with unity -- let's say $\mathbb{Z}$ just for the sake of argument -- and consider $\mathbb{Z} \oplus \mathbb{Z}$. This is a ring with unit element $(1,1)$. Now consider the subring consisting of all elements of the form $(a,0)$. You can take it from there...

Answer 2

Consider the subrng of $\,2\times 2\,$ matrices having form $\ \left[\begin{array}{}r & 0\\ 0 & 0\end{array}\right]$