All rings considered are unital and commutative.
Intro Direct products of domains are reduced rings. The opposite is not true but I must admit I have troubles finding counter-examples. It holds that reduced rings are subdirect products of domains. But this is not that useful as many interesting ring-theoretic properties are not closed under subalgebras (e.g. being Artinian).
Question What are some general, ring-theoretic properties that characterize when a reduced ring is a finite direct product of domains? Are there at least some special cases (rings of small dimension, with zero socle, zero radical...) for which this has some nice characterization?
I do not need this for a specific purpose, and I am not sure whether this has some good, clear answer. It is just a question that is intrinsically interesting to me.
I am not sure how to start. An obvious necessary condition is that socle of the ring is zero (because domains that are not fields have zero socle). But I can't prove that this is sufficient, neither can I find a counterexample.
EDIT: Thanks Eric for pointing out that direct sum of fields is counterexample. As a field is (the only) example of domain that has non-zero socle.