Let $R$ be a (commutative) Noetherian ring, and let $\{I_n\}$ be a chain of ideals of $R$. As we know, the Noetherian property of $R$ indicates that $\{I_n\}$ satisfies the ascending chain condition, namely, there exists some $m \in \mathbb{N}$ such that $$ I_{m - 1} \subseteq I_m = I_{m + k} \text{ for all } k \in \mathbb{N}. $$ I was wondering given a specific $R$ and specific chain $\{I_n\}$ whether there is any way to determine what this integer $m$ is. At this moment, I cannot construct even one explicit non-trivial example (in particular, I was trying with rings of single variable/multivariable polynomials/power series). Any examples, or any indication towards some general theory or reference material will be highly appreciated.
2026-03-29 19:09:57.1774811397
Chain of ideals in a Noetherian ring
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