ON POWERS OF IDEALS $\left(I^m J^n\right)_{(m, n) \in \mathrm{N}^2}$ AND $\sigma$-SPORADIC PRIME DIVISORS OF A PAIR OF IDEALS $(I, J)$

Abstract

Let $I$ and $J$ be two non-zero ideals of a Noetherian ring $A$. Let $\sigma$ be a semi-prime operation on the set of all ideals of $A$. Let $m_0$ and $n_0$ be two fixed positive integers, and

$$
\mathrm{E}_{m_0, n_0}=\left\{\left(\beta m_0, \beta n_0\right) / \beta \in \mathrm{N}\right\} .
$$

In first time, we show that:
(a) for $e \in \mathrm{E}_{m_0, n_0} \backslash\{(0,0)\}, \quad\left\{\sigma\left(\mathrm{K}^{e+t}\right): \sigma\left(\mathrm{K}^t\right)\right\}_{t \in \mathrm{E}_{m_0, n_0}}$ is an increasing sequence and stabilizes at $\mathrm{K}_\sigma^e$,
(b) the sequence $\left(\operatorname{Ass}\left(A / \sigma\left(\mathrm{K}^e\right)\right)\right)_{e \in \mathrm{E}_{m_0, n_0}}$ is increasing and stabilizes at $A_\sigma(I, J)=A_\sigma(\mathrm{K})$ under certain conditions for $e$ large enough.

First, we show

$$
S_{\ell}^\sigma(I, J)=\operatorname{Ass}\left(A / \sigma\left(\mathrm{K}^{\ell}\right)\right)-A_\sigma(\mathrm{K}),
$$

for $\ell$ small enough in $\mathrm{E}_{m_0, n_0}$. The elements of $S_{\ell}^\sigma(I, J)$ are called $\sigma$-sporadic prime divisors of $(I, J)$.

We give some properties on the set $S_{\ell}^\sigma(I, J)$.
To finish, we conclude this paper with the equality $\mathrm{K}_\sigma^{\ell}=\left[\sigma\left(\mathrm{K}^{\ell}\right)\right]_{\Delta}$ under certain conditions, where $L_{\Delta}$ is the $\Delta$-closure of the ideal $L$ of $A$.

Received: August 12, 2025
Revised: September 20, 2025
Accepted: October 18, 2025