= Noetherianity of a formal power series ring
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A <formal power series ring> $R[[x]]$ is <Noetherian> if and only if $R$ is <Noetherian>. In the forward direction, use the <quotient ring> $R[[x]]/(x)\cong R$. In the reverse direction, for an <ideal> $I$ consider the ascending sequence of <ideals> of $R$ formed by the coefficient of $x^n$ in $I\cap x^nR[[x]]$. Stabilization and finite generation give a finite list of series that generate $I$ by successive coefficient cancellation.
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