Let $a$ and $b$ be two elements in a commutative ring $R$ and $(a, b) = R$, show that $(a^m, b^n) = R$ for any positive integers $m$ and $n$.

Question

I stumbled across a question that I have no idea how to start.

I know the questions asking to show that the multiples of $a$ and $b$ as an ordered pair make still make the whole ring.

Any sort of hints or suggestions to start? I don't really want an complete solution, just some sort of hint to start the question.

Thank you!

Answer 1

Write $1$ as a linear combination of $a$ and $b$. Then raise both sides to the $m+n$ power.

Answer 2

The condition $(a,b) = R$ is equivalent to saying that $\overline{b}$ is invertible in $R/(a)$. Now, what can you say about $\overline{b}^m$?