Note on linearly equivalent ideal topologies over Noetherian modules

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 $\mathfrak{a}$ of $R,$ the $\mathfrak{a}$-symbolic topology on $N$ is linearly equivalent to the $\mathfrak{a}$-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 $\dim N\le 1$.

5 pages

References in corpus (2)

• Symbolic powers of ideals and their topology over a module
• Asymptotic behaviour of integral closures, quintasymptotic primes and ideal topologies