Does every Integral Domain satisfy Decending chain condition? Or there should be a stronger restriction?
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 be in D. consider the chain so there exists an n > 0 such that and hence for some d in D. but then
and thus because and D is a domain. so every non-zero element of D is invertible, i.e. D is a field.
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.