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The exact question is:
Is it possible that a ring with unity may simultaneously contain subrings isomorphic to $\mathbb{Z}_n$ and $\mathbb{Z}_m$, where $n \ne m$?
The answer says it's possible and says $\mathbb{Z}_2 \times \mathbb{Z}_3$ has a subring isomorphic to $\mathbb{Z}_2$ and one isomorphic to $\mathbb{Z}_3$. I'm just trying to understand the problem better, but I can't see which two subrings are isomorphic. Could someone explain this solution to me? Thank you.
If you are working in the category of rings, yes, as shown in another answer.
But if you are working in the category of rings with unity, that is, if you insist on subrings having the same unity as the ring in which they are contained, then you have a problem.
In fact, in $\mathbb{Z}_n$ you have $n \cdot 1 = 0$ (where $1$ is the unity, and $n \cdot 1$ denotes the $n$-th multiple of $1$), and $k \cdot 1 \ne 0$ for $0 < k < n$. In $\mathbb{Z}_m$ you have $m \cdot 1 = 0$, and $k \cdot 1 \ne 0$ for $0 < k < m$.
If $m < n$ or $n < m$ we obtain a contradiction. So $m = n$.
The two are $\mathbb{Z}_2\times \{0\}$ and $\{0\}\times \mathbb{Z}_3$ with the most natural isomorphism $$\mathbb{Z}_2\times \{0\}\cong \mathbb{Z}_2,$$ and analogously for the other subring.
