In a Euclidean ring $R$, prove $(a) ⊆ (b) \iff b|a$

Question

Let $a, b$ be elements of a Euclidean ring $R$. Prove that $$(a) \subseteq (b) \iff b \;\text{divides}\;a.$$

I have no clue how to even start this. Any help would be great, thank you in advance!

Answer 1

$\impliedby$: Suppose $a=bk$. Let $ x \in (a)$. Then $$x=ta=t(bk)=(tk)b \in (b)$$

$\implies:$ Suppose $(a) \subset (b)$. Then $a \in (b)$ implies $a=bk$, concluding $b|a$