Does every Integral Domain satisfy Decending chain condition? Or there should be a stronger restriction?

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- Aug 18th 2009, 12:15 AMynjDecending chain condition and Integral Domain
Does every Integral Domain satisfy Decending chain condition? Or there should be a stronger restriction?

- Aug 18th 2009, 12:19 AMNonCommAlg
- Aug 18th 2009, 12:25 AMynj
yes

- Aug 18th 2009, 12:26 AMNonCommAlg
- Aug 18th 2009, 12:29 AMynj
well..then something wrong with my text..

Can any one give a sufficiency condition about DCC on ideals?? - Aug 18th 2009, 12:36 AMNonCommAlg
actually an integral domain satisfies the descending chain condition if and only if it's a field. the proof is very easy! one side is clear. now suppose an integral domain D satisfies the descending

chain condition and let $\displaystyle x \neq 0$ be in D. consider the chain $\displaystyle <x> \supseteq <x^2> \supseteq <x^3> \supseteq \cdots .$ so there exists an n > 0 such that $\displaystyle <x^n>=<x^{n+1}>$ and hence $\displaystyle x^n=dx^{n+1},$ for some d in D. but then

$\displaystyle x^n(dx -1)=0$ and thus $\displaystyle dx=1$ because $\displaystyle x \neq 0$ and D is a domain. so every non-zero element of D is invertible, i.e. D is a field. - Aug 18th 2009, 12:41 AMynj
- Aug 18th 2009, 12:47 AMNonCommAlg
descending chain condition is for all ideals, proper or not proper. so, you can say this: an integral domain satisfies the descending chain condition if and only if it's a field.

by the way rings satisfying the descending chain condition (on their ideals) have a name. they're called Artinian.