BIFILTRATIONS ON A RING AND ASYMPTOTIC $\sigma$-PRIME DIVISOR

Abstract

Let $I$ and $J$ be two nonzero ideals of a Noetherian ring $A$. We put $\mathbb{E}_{m_0, n_0}=\left\{\left(\beta m_0, \beta n_0\right) / \beta \in \mathbb{N}\right\}$, where $m_0$ and $n_0$ are two fixed positive integers. For all $e=\left(e_1, e_2\right) \in \mathbb{E}_{m_0, n_0}$, we put $K^e-I^{e_1} J^{e_2}$. Let $s$ be a semi-prime operation on the set of all ideals of $A$. Then we show that $\left\{\sigma\left(K^{e+t}\right): \sigma\left(K^t\right)\right\}_{t \in \mathbb{B}_{m_0, n_0}}$ is increasing, so it stabilizes. Putting $K_\sigma^e=\sigma\left(K^{e+t}\right): \sigma\left(K^{\prime}\right)$ for $t$ large enough in $\mathbb{E}_{m_0, n_0}$, we prove that, if $\ell=\left(\ell_1, \ell_2\right) \in \mathbb{E}_{m_0, n_0}$ with $\ell_1, \ell_2 \geq 1$, then for all $e \in \mathbb{E}_{m_0, n_0}, A s s\left(A / K_\sigma^{\ell}\right)$ is contained in $A s s\left(A / K_\sigma^{\ell+e}\right)$. Finally, we show that $\left(K_\sigma^e\right)_{e \in \mathbb{E}_{m_0, m_0}}$ is a bifiltration on the ring $A$.

Received: January 2, 2023
Revised: January 29, 2023
 Accepted: February 2, 2023