Let $R$ be a commutative ring with unity different from $0$, and $I$ be a decomposable ideal of $R$, i.e. $$I=\mathfrak{q}_1\cap \mathfrak{q}_2 \cap \cdots \cap \mathfrak{q}_r $$ where (1) $\mathfrak{q}_i$ are primary ideals; (2) $rad(\mathfrak{q}_i)$ are all distinct; (3) No $\mathfrak{q}_i$ contains intersection of other $\mathfrak{q}_j$'s.
In this situation, it is well-known that
(1) $\mathfrak{p}_i$ are uniquely determined (independent of whether $R$ is noetherian or not).
(2) $\mathfrak{p}_i$ are precisely the prime ideals appearing in the set $\{ rad(I:x) \,\, | \,\, x\in R\}$.
If $R$ is noetherian, we can say more (see Prop.2.10(iii), page 21 for proof):
(2') $\mathfrak{p}_i$ are precisely the prime ideals appearing in the set $\{ (I:x) \,\, | \,\, x\in R\}$.
Q. Can't we conclude (2') if $R$ is not Noetherian?