Why is it true that if a ring and subring are both integral domains, then their unity elements are equal?
I have seen this statement as a property of integral domain subrings. But I cannot figure out why is it specifically for an integral domain? How no zero divisors axiom is related to unity? (or commutativity)
So the question is: if $D$ is an integral domain, and $R\subseteq D$ is a subring that is also an integral domain on its own right, must the identity of $R$ be the same as the identity of $D$?
Yes, it must. (Note that $R$ does not have any zero divisors anyway, so the only question is whether it has an identity different from its zero).
Note that over an integral domain $D$, the only elements $x$ such that $x^2=x$ are $x=1_D$ and $x=0_D$. Because $x^2=x$ means $x^2-x=0_D$, hence $(x-1_D)x=0_D$, so either $x=0_D$ or $x-1_D=0_D$.
So if $R\subseteq D$ is a subring with unity, then $1_R\in R$ satisfies $1_R1_R=1_R$. But this also holds in $D$. So that means that either $1_R=0_D$ (so $R$ is the zero ring), or $1_R=1_D$. Thus, if $R$ is not the zero subring, then the unity of $R$ is the same as the unity of $D$.
So either $R$ is the zero ring (and not an integral domain), or it has the same unity as $D$ (and therefore must be an integral domain, since it has a unity but no zero divisors).
By the way: this holds over a ring with unity and no zero divisors, even if it is not commutative. So the statement holds over domains (nonzero rings with unity that have no left or right zero divisors), not just integral domains.