Showing that $\mathbb Z_3$ is not a subring of $\mathbb Z_5$ I'm having problems in identifying that $\mathbb Z_3$ is a subset of $\mathbb Z_5$. Because:
$$\mathbb Z_3 = \{0,1,2\}$$
$$\mathbb Z_5 = \{0,1,2,3,4\}$$
But in $\mathbb Z_3$, $0$ is actually a set of all the numbers congruent to $0 \mbox{ mod} 3$. So 
$$0_3 =\{\cdots, 3, 6, 9, \cdots\}\\1_3 = \{\cdots, 4, 7, 10, \cdots\}\\2_3 = \{\cdots, 5, 8, 11, \cdots \}$$
Now, for $\mathbb Z_5$ we have:
$$0_5 = \{\cdots, 5, 10, 15\cdots\}\\1_5 = \{\cdots, 6, 11, 16\cdots\}\\2_5 = \{\cdots, 7, 12, 17\cdots\}\\3_5 = \{\cdots, 8, 13, 18\cdots\}\\4_5 = \{\cdots, 9, 14, 19\cdots\}$$
but none of the sets in $\mathbb Z_3$ is in $\mathbb Z_5$. So could I say that $\mathbb Z_3$ is not a subring of $\mathbb Z_5$ because $\mathbb Z_3$ is not even a subset of $\mathbb Z_5$? Is my reasoning correct?
Could you give an example of $\mathbb Z_n$ and $\mathbb Z_k$ such that one is subring of another?
 A: Even if you could see $\mathbb{Z}_3$ as a subset of $\mathbb{Z}_5$ (which you cannot in any natural way), it could not be a subring, because a subring is in particular a subgroup for the additive structure: Lagrange's theorem disallows it, because $3\nmid 5$.
A: It depends on your definition of $\mathbb{Z}_5$ and $\mathbb{Z}_3$ whether $\mathbb{Z}_3$ can be considered as a subset of $\mathbb{Z}_5$ or not. If you consider $\mathbb{Z}_3$ and $\mathbb{Z}_5$ as quotients of $\mathbb{Z}$ by the relevant congruence relation, then as you have noted, $\mathbb{Z}_3$ isn't even a subset of $\mathbb{Z}_5$. However, one could have defined $\mathbb{Z}_n$ to be the set $\{0, 1, \ldots, n \}$ with the operations of addition and multiplication modulo $n$ and then $\mathbb{Z}_n$ can be considered a subset of $\mathbb{Z}_m$ for $n < m$. Assuming the second definition, $\mathbb{Z}_3$ will be a subset of $\mathbb{Z}_5$ but won't be a subring of $\mathbb{Z}_5$.
If you are familiar already with the notion of rings homomorphisms and isomorphisms, the two different constructions described in the first paragraph result in isomorphic rings and so from the point of view of ring theory they can be considered the same (after an identification - the isomorphism - is applied). A better statement for your problem that doesn't take into account the specific way in which $\mathbb{Z}_n$ were constructed in the first place is to show that there does not exist an injective homomorphism of rings $\varphi \colon \mathbb{Z}_3 \rightarrow \mathbb{Z}_5$. This makes sense no matter how the $\mathbb{Z}_n$'s were constructed and takes into account only the ring theoretic properties of the $\mathbb{Z}_n$ and not the (irrelevant for this discussion) set theoretic properties. If such an injective homomorphism of ring would exist, the image $\varphi(\mathbb{Z}_3)$ would be both a subset and a subring of $\mathbb{Z}_5$ allowing us to identify $\mathbb{Z}_3$ as a "true" subring of $\mathbb{Z}_5$. 
A: Note that the addition operations are different, $mod3$ and $mod5$. When we study subrings, it is necessary that the operation be the same. eg. take $1$ out of $\mathbb{Z_3}$ and add it thrice. In the ring $\mathbb{Z_3}$, $1+1+1=3.1=0$ but in $\mathbb{Z_5}$, $1+1+1=3.1=3\neq 0$.
Hence we conclude that $\mathbb{Z_n}$ is not contained as subring in $\mathbb{Z_k}$ if $k\neq n$.
