Property of GCD in ring

Question

Let D be a domain and $\emptyset \subset A \subseteq D^*$

Show that CD(A)={$d\in D$ | $(A)\subseteq (d)$}

I know that I'll need to show both containments to show that the two statements are equivalent but I'm not sure how to use the definitions exactly to complete the proof.

Common Divisors CD(A)- if $d \in CD(A)$ then d|a for all $a \in A$

I also know that if $(A) \subseteq (d)$ then d|A

Any advice would be great!

Answer 1

You say "$(A) \subseteq (d)$ then $d\mid A$". That is not accurate (a number cannot divide a set), what you should say is "$(A) \subseteq (d)$ if and only if $d\mid a$ for all $a\in A$". In other words $(A) \subseteq (d)$ if and only if $d\in CD(A)$. That's your proof.