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ASYMPTOTIC STABILITY OF $A S S_A\left(\frac{A}{I_n}\right)$ FOR STRONGLY NOETHERIAN FILTRATION $f=\left(I_n\right)_{n \in \mathbb{N}}$
Keywords:
Noetherian ring, filtration, associated prime ideals, superficial elementDOI:
https://doi.org/10.17654/0972555525014Abstract
Let $A$ be a Noetherian ring, and $f=\left(I_n\right)_{n \in \mathbb{N}}$ be a strongly Noetherian filtration on the ring $A$. Then we generalize the Artin-Rees Lemma to strongly Noetherian filtrations. This allows us to show that if the ideal $I_1$ contains an $f$-superficial element of order one which is regular, then the sequence $\left(A S S_A\left(\frac{A}{I_n}\right)\right)_{n \geq 1}$ stabilizes.
Received: January 2, 2025
Revised: February 18, 2025
Accepted: March 10, 2025
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