Ascending filtered-graded transfer of Noetherianity ($\mathrm{gr}A\text{ left Noetherian}⟹A\text{ left Noetherian}$)

For an exhaustive nonnegative increasing multiplicative filtration, each left ideal $I$ induces a homogeneous ideal $\mathrm{gr}I$ in the associated graded ring. If that graded ring is a left Noetherian ring, lift a finite homogeneous generating set of $\mathrm{gr}I$ 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.