What does it mean for an ideal in the ring of integers to divide another ideal in the ring of integers?

Question

How does the definition of integer divisibility carry over to ideals?

Answer 1

In the case of principal ideal domains, an ideal $I$ divides an ideal $J$ of a PID $R$ if and only if $J \subseteq I$. The intuition here is that, if $J = (a)$ and $I = (b)$,

$$ J \subseteq I \iff a \in (b) \iff a = bc \text{ for some $c \in R$} \iff b | a. $$

which matches with the idea of divisibility we already have: $(b)$ divides $(a)$ if and only if $b$ divides $a$. More generally, this concept can be carried over to Dedekind domains (in which non trivial ideals can be decomposed as product of powers of prime ideals, similar to the integers): if $I = \mathfrak{p_1}^{\alpha_1} \cdots \mathfrak{p_n}^{\alpha_n}$ and $J = \mathfrak{q_1}^{\beta_1} \cdots \mathfrak{q_n}^{\beta_n}$ with $\mathfrak{p}_i, \mathfrak{q}_j$ prime then $I$ divides $J$ if and only if each $\mathfrak{p}_i$ coincides with some $\mathfrak{q}_j$ and $\alpha_i \leq \beta_j$.