I know this is probably not that hard but I don't know how to properly approach this.
So I am asked to give an example of a ring fulfilling the properties in the title of the question. Now I know that a commutative Noetherian ring is Artinian iff it has finite and discrete spectrum, so that rules out all of those. I don't really know how to go on from there (also because I think I'm not properly understanding how to "work" with open neighbourhoods in the Zariski topology yet). Can someone maybe give me a hint?