My book has proven this:
Every ring with unity has a subring isomorphic to either $\mathbb{Z}$ or $\mathbb{Z}_n$.
The $\mathbb{Z}_n$ case arises if the parent ring has characteristic $r>0$.
I have this question:
Is it possible for a ring with characteristic $r>0$ to have two subrings isomorphic to $\mathbb{Z}_n$ and $\mathbb{Z}_m$ with $n \neq m$?
I believe so, yes. Consider $\mathbb{Z_8}$. clearly $\mathbb{Z_6} \subset \mathbb{Z_8}$ and $\mathbb{Z_6} \cong \mathbb{Z_6}$, and $\mathbb{Z_4} \subset \mathbb{Z_8}$ and $\mathbb{Z_4} \cong \mathbb{Z_4}$.
My question is how to prove this.