Let $R$ be a Cohen-Macaulay ring with no non-trivial idempotent element. Then is it true that there is a Noetherian domain $S$ such that $R\cong S/I$ for some ideal $I$ of $S$ ?
If this is not true in general, what if we also assume $R$ has finite Krull-dimension ?
MOTIVATION: A theorem of Hungerford https://msp.org/pjm/1968/25-3/pjm-v25-n3-p11-p.pdf says that every PIR is a homomorphic image of a direct product of finitely many PID s. So I was thinking about the same question for special kind of Noetherian rings