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Is there any characterization for a commutative ring to be a quotient of a Dedekind domain?

bastini
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    Let $R$ be a dedekind domain and $I$ and ideal, and let $\prod \mathfrak{p}_i^{e_i}$ be $I$'s factorization. Then $R/I = \prod R/\mathfrak{p}_i^{e_i}$ by the CRT, and each $R/\mathfrak{p}_i^{e_i}$ is an artin ring. So $R/I$ is a finite direct product of artin rings. The question is which such finite products can occur. – benblumsmith Jul 11 '16 at 16:21
  • Make that local artin rings. – benblumsmith Jul 11 '16 at 16:40
  • Another major condition is that the Zariski tangent space of each $R/\mathfrak{p}_i^{e_i}$ is one dimensional. We also get a restriction on the residue fields from http://mathoverflow.net/questions/176117/existence-of-a-ring-with-specified-residue-fields/176194#176194 . – David E Speyer Jul 11 '16 at 17:34
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    above should say "one or zero" dimensional. – David E Speyer Jul 11 '16 at 18:46

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