The characteristic of a subdomain is the same as the integral domain in which it is contained? Perhaps I have a misunderstanding of what a subdomain and an integral domain are, but I'm having a hard time figuring this out.
I'm asked to show that the characteristic of a subdomain is the same as the characteristic of the integral domain in which it is contained.
What was tying me up is: $\mathbb Z_7$ is an integral domain. $\mathbb Z_3$ is also an integral domain, and every element in $\mathbb Z_3$ is contained in $\mathbb Z_7$, so isn't $\mathbb Z_3$ a subdomain of $\mathbb Z_7$?
I assume it's probably fairly simple (a misunderstanding of a definition or something), but what am I missing here?
(Edit: I, apparently, had a lapse in brain functioning which resulted in a pretty bad misunderstanding of subrings. Once this was fixed, the proof came naturally.)

A subdomain $R \subseteq S$ is first of all a subring, i.e. the inclusion $i : R \hookrightarrow S$ is a ring homomorphism. Since ring homomorphisms send $1$ to $1$ by definition, $R$ and $S$ must have the same unity element $1$, so $p \cdot 1 = 0$ in $R$ iff $p \cdot 1 = 0$ in $S$, which shows that $R$ and $S$ have the same characteristic. 
As for your example, $\mathbb{Z}_3$ and $\mathbb{Z}_7$ do not have the same unity element, and in fact not only is $\mathbb{Z}_3$ is not a subset of $\mathbb{Z}_7$, there is no injective ring homomorphism from $\mathbb{Z}_3$ to $\mathbb{Z}_7$.
