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 $N\isin\mathbb{N}\,$ such that $I_N = I_{N+1} = \cdot\cdot\cdot\,$ ) is a field.

Let $a\isin R\,$ . 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 $a^N = b a^{N+1}\,$ for some nonunit $b\isin R\,$ .

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

$1=ab\,$ because $a^N\ne 0\,$ and $R\,$ is an integral domain.

Therefore $a\,$ is a unit.

Main Page : Abstract Algebra : Rings

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