If spectrum of a ring is connected, then the ring is integral domain If the spectrum of a ring is connected, then will the ring be an integral domain?
The spectrum is set of all prime ideals. Given the Zariski topology, how is it that connectedness of the set of prime ideals imply that there are no zero divisors? I think the nilradical of the ring is a subset of the spectrum of the ring. I am unable to start the problem. Any hints? Thanks beforehand.

There are lots of commutative rings that are connected but are not domains.
Local rings that are not domains are a very large class of that type.
But there are non-local examples too, like $\mathbb Z[x]/(x^2-1)$.
