judge if nilradical equals jacobson radical
1)a noetherian ring that is not a artin ring.
2)a local integral domain that is not a field.
3)a integral domain with only finite number of maximal ideals that is not a field.
4)a ring that only has finite number of prime ideals in which it's not the case that all primes are maximal.
I think 1)should be wrong but example like $\mathbf{Z}$ does not work...
and 3) is a special case of 2), I mean if 3) is wrong then 2) is also wrong..
and for the rest, I have no idea..
(1) In a Noetherian ring nilradical can be equal to Jacobson radical, even if it is not a Artin ring. $\mathbb{Z}$ is an example. In fact any Jacobson ring has this property.
(2) is not true, because in this case the nilradical is the zero ideal, but Jacobson radical is a non-zero ideal (since it is not a field). Take any Noetherian local domain which is a DVR.
(3) is not true in general because of (2).
(4) is not true in general again because of (2).