Could anyone give me a reference for the proof of “ In a commutative artinian ring, every maximal ideal is a minimal prime ideal and every minimal prime ideal is a maximal ideal” ?
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Atiyah's introduction to commutative algebra:commutative ring is Artinian ring iff it is Noetherian ring and the Krull dimension is zero. – Jian Jan 26 '18 at 11:07
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How this is relevant ? – Math90 Jan 26 '18 at 11:20
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you can find the definition of Krull dimension in Wikipedia. – Jian Jan 26 '18 at 11:24
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What is your definition of Artinian ring? – lhf Jan 26 '18 at 11:36
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What is your work on the subject ? Is the second part more difficult ? – Jean Marie Jan 26 '18 at 11:52
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@lhf The ring which satisfies d.c.c – Math90 Jan 26 '18 at 12:28
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@jean Marie I want to prove the both part, and I don’t know how to use krull dimension for this – Math90 Jan 26 '18 at 12:30
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1See https://math.stackexchange.com/questions/27145/why-are-artinian-rings-of-krull-dimension-0 – lhf Jan 26 '18 at 12:47