TITLE: Why are Artinian rings of Krull dimension 0?
QUESTION [15 upvotes]: Why are Artinian rings of Krull dimension 0?
As in the example of $\mathbb{Z}/(6)$, the ideal $\mathbb{Z}/(2)$ is prime, I think. So, Artinian rings may contain prime ideals. But why does not the primes ideal contain other prime ideals properly?
Another question is that, when a ring is of Krull dimension 0, is it necessarily Artinian?
Thanks.

REPLY [14 votes]: HINT $\ $ By factoring out a prime ideal it reduces to showing that an Artinian domain is a field, which follows immediately by DCC.
This method of factoring out by a prime ideal to reduce from rings to domains is a ubiquitous algebraic problem solving technique. Indeed, the eminent algebraist Irving Kaplansky explicitly pauses to mention this method (in a less trivial context) in his classic textbook Commutative Rings. Follow the above link for an excerpt. Not only was Kap a great algebraist but also a great expositor - a rare combination. I wholeheartedly recommend his expositions, where I recall learning many beautiful mathematical ideas.