Specific subring with non-zero unity element 
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

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...
