Confusing assumption in exercise: $a^n=b^n$, $a^m=b^m$ implies $a=b$ in [integral domain, ring with no zero divisors]

Confusing assumption in exercise: $a^n=b^n$, $a^m=b^m$ implies $a=b$ in
[integral domain, ring with no zero divisors]

My textbook gives the following exercise: let $a$ and $b$ be elements in
an integral domain $R$. If $(m,n)=1$, show that $a^n=b^n$ and $a^m=b^m$
implies $a=b$. Then it asks the same thing, with 'integral domain'
replaced by 'ring where $xy=0$ implies $x=0$ or $y=0$'. The proofs given
in both cases are different, though I don't understand why, and I don't
understand why the conditions on the ring are necessary.
Here is an attempt at proving this for an arbitrary ring: since $(m,n)=1$
we have $nu+mv=1$ for some $u,v$. Then
$a=a^{nu+mv}=a^{n^u}a^{m^{v}}=b^{n^u}b^{m^v}=b^{nu+mv}=b$. This is the
proof they give for the case where $R$ is an integral domain, but it's
different from the one where $R$ is not. I don't understand where I'm
using this assumption here.