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.)

Notice $1+1+1=0$ in $Z_3$ but $1+1+1\color{Red}{\ne}0$ in $Z_7$. Just because the symbols used to represent the elements of $Z_3$ are a subset of the symbols used to represent the elements of $Z_7$ doesn't mean that one structure sits inside the other structure. Not even remotely close. For a subset to be a subring its operations must be the same as those in the ring it sits inside.
If $A\to B$ is any homomorphism of unital rings, we have ${\rm char}(B)\mid{\rm char}(A)$. Can you prove this? Consider the fact that the element $\underbrace{1_B+\cdots+1_B}_{{\rm char}(A)}\in B$ is the image of $\underbrace{1_A+\cdots+1_A}_{{\rm char}(A)}=0_A$.
In particular if $B$ is a domain then its characteristic is either zero or a prime number (can you prove this fact?). If ${\rm char}(B)=p$ then $A$ is a domain and its characteristic is a multiple of $p$, so the characteristic of $A$ must also be $p$. I'll let you figure out the characteristic zero case.
