Let $A$ a integral domain and let $\mho(A)$ the set of all non-zero ideals. Show that the set of all principal ideals is an equivalence class of the relation $\sim$ that we can noted by $[A]$.
The equivalence relation on $\mho(A)$ is simply $I \sim J$ if there exists $\alpha, \beta \in A-{0}$ such that $\alpha I = \beta J$.
This equivalence class seems a bit strange for me. Do I have to prove that $\{ all-principal-ideals-in-A \}$ $= \{I \in \mho(A) : I \sim I_p\}$, where $I_p$ is a principal ideal?
I think the point is that you want to show every principal ideal is in the same equivalence class as $A$. This is true because given a principal ideal $I=(\alpha)$, we have $\alpha A=1I$, so $I\sim A$. Therefore the equivalence class of all principal ideals is $[A]$, the equivalence class of $A$.