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..

• Is this your homework? It doesn't seem you are spending enough to time to think about it before asking. – Manos Dec 21 '14 at 1:29

(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.
• yeah, it's true, but can you give me some concrete examples of (1) (2) (4) that the two radical are not equal? and for(3) is $k\times k$ a field? I think they may not be integral domain, like $(0,1)\times (1,0)$ is $(0,0)$ in $Z_2 \times Z_2$ – annimal Dec 21 '14 at 2:51
• @annimal: Ah..!! Sorry. in (3) I missed the "integral domain" part. I'll edit my answer. and you are right $k \times k$ is not an integral domain. – Krish Dec 21 '14 at 3:01