Let $D$ be a Prüfer domain. I am looking for equivalent condition on an ideal $I$ of $D $ under which $D/I $ is an indecomposable ring.



Indecomposable quotient of Prüfer domains
If $D$ is a [Dedekind domain](https://en.wikipedia.org/wiki/Dedekind_domain) and $I$ an ideal of $D$, the quotient ring $D/I$ is directly [indecomposable](https://en.wikipedia.org/wiki/Indecomposable_module), or equivalently, [connected](https://en.wikipedia.org/wiki/Connected_ring), if and only if $I$ is [primary](https://en.wikipedia.org/wiki/Primary_ideal).


This remark generalizes to [Prüfer domain](https://en.wikipedia.org/wiki/Pr%C3%BCfer_domain) of *finite character*, i.e., Prüfer domains in which any non-zero element is contained but in a finite number of maximal ideals.



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> > **Claim.** Let $D$ be a Prüfer domain of finite character and let $I$ be an ideal of $D$. Then the quotient ring $D/I$ is directly indecomposable if and only if $I$ is quasi-primary.
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An ideal is said to be *quasi-primary* in the sense of Laszlo Fuchs and Edward Mosteig if its radical is prime.



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> > *Proof of the Claim.* By [1, Theorem 5.7], every non-zero ideal $I$ of $D$ is the intersection of finitely many pairwise [comaximal](https://en.wikipedia.org/wiki/Ideal_(ring_theory)) quasi-primary ideals uniquely determined by $I$.
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[1] L. Fuchs and E. Mosteig, ["Ideal theory in Prüfer domains
—An unconventional approach"](https://ac.els-cdn.com/S0021869302000406/1-s2.0-S0021869302000406-main.pdf?_tid=8342ca68-014e-11e8-bf99-00000aab0f27&acdnat=1516829916_3b967051f580b2fa2e15d2d45b51e946), 2002. 


