Let $A$ be an integral domain with $A$ a commutative ring. Let $m \in A$ be such that $(m)$ is the principal ideal of the intersection of the principal ideal $(a)$ and the principal ideal $(b)$. We want to show that for all elements $c$ which divides $a$ and divides $b$ it can also be divisible by $m$.
I would like to know if what I have written is correct.
We have $a|c$ and $b|c$ which means that $(c) \subset (a)$ and $(c) \subset (b)$. By that we have that $(c) \subset (a) \cap (b)$ which implies that $m|c$.
I would appreciate if you could inform me if my approach is good or if there is something that I should revise.