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.)