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?

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    $\begingroup$ Isn't the spectrum of a discrete valuation ring not discrete because the closure of the zero ideal is more than just the zero ideal? When the zero ideal is prime, and there are primes properly containing it, doesn't its closure include more than just itself? I'm probably not much better at Zariski topology either... I haven't thought about this in a long time. But I thought chains of primes interfered with discreteness. $\endgroup$ – rschwieb Jun 3 at 20:11
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    $\begingroup$ Hint: DVR. (Filler text) $\endgroup$ – KReiser Jun 3 at 20:12

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