AAR9

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Show that an integral domain R\, with a descending chain condition (if I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq \cdot \cdot \cdot \, is a descending chain of ideals, then there exists N\in {\mathbb  {N}}\, such that I_{N}=I_{{N+1}}=\cdot \cdot \cdot \,) is a field.


Let a\in R\,. Then there is a descending chain (a)\supset (a^{2})\supset (a^{3})\supset ...\,.

Then (a^{N})=(a^{{N+1}})=...\, for some N\,.

This implies a^{N}=ba^{{N+1}}\, for some nonunit b\in R\,.

a^{N}(1-ab)=0\,

1=ab\, because a^{N}\neq 0\, and R\, is an integral domain.

Therefore a\, is a unit.



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