# Stability in graded rings associated with commutative augmented rings

## TL;DR

This paper proves that the sequence of quotients of powers of the augmentation ideal in a commutative augmented ring stabilizes, leading to stability in the associated graded ring structure.

## Contribution

It establishes the stability of the sequence of quotients of powers of the augmentation ideal in commutative augmented rings, a new result in ring theory.

## Key findings

- Sequence $ ext{I}^n/ ext{I}^{n+1}$ becomes stationary up to isomorphism
- Stability in the associated graded ring of $A$ along $I$
- Provides a foundational result for understanding the structure of augmented rings

## Abstract

Let $A$ be a commutative augmented ring and $I$ be its augmentation ideal. This paper shows that the sequence $\{I^n/I^{n+1}\}$ becomes stationary up to isomorphism. The result yields stability in the associated graded ring of $A$ along $I$.