= Ascending filtered-graded transfer of Noetherianity
{title2=$\operatorname{gr}A\text{ left Noetherian}\implies A\text{ left Noetherian}$}
For an exhaustive nonnegative increasing multiplicative <filtration>, each <left ideal> $I$ induces a homogeneous <ideal> $\operatorname{gr}I$ in the <associated graded ring>. If that graded ring is a <left Noetherian ring>, lift a finite homogeneous generating set of $\operatorname{gr}I$ to elements of $I$. For any element of $I$, subtract a combination of these lifts with the same leading symbol; the remainder has strictly lower degree. Induction on the nonnegative degree proves the lifts generate $I$. There is no completion hypothesis because the cancellation terminates in finitely many steps. The analogous result holds for <right ideals>.
Back to article page
