Note on linearly equivalent ideal topologies over Noetherian modules

TL;DR

This paper characterizes when the symbolic and adic topologies on a finitely generated module over a Noetherian ring are linearly equivalent, linking it to the structure of associated primes and the module's dimension.

Contribution

It provides a precise criterion for the linear equivalence of symbolic and adic topologies on modules over Noetherian rings, connecting it to prime spectra and module dimension.

Findings

Symbolic and adic topologies are linearly equivalent iff each localization has a single associated prime.

The equivalence holds if and only if the module's dimension is at most one.

The result characterizes the topological equivalence in terms of prime ideal structures.

Abstract

Let $R$ be a commutative Noetherian ring, and let $N$ be a non-zero finitely generated $R$-module. In this paper, the main result asserts that for any $N$-proper ideal $\frak a$ of $R,$ the $\frak a$-symbolic topology on $N$ is linearly equivalent to the $\frak a$-adic topology on $N$ if and only if, for every $\frak p\in \Supp(N)$, $\Ass_{R_{\mathfrak {p} }}N_{\mathfrak {p}}$ consists of a single prime ideal and $\dim N\leq 1$.