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CLOSURE OPERATION EXTENDED TO REES RING AND ASYMPTOTIC PRIME DIVISORS
Keywords:
Rees ring, closure operation, prime divisorsDOI:
https://doi.org/10.17654/0972555524018Abstract
Let $A$ be a Noetherian ring and $I$ be a nonzero ideal of $A$. Let $\mathcal{R}(A, I)$ be the generalized Rees ring of the ideal $I$. Let $\sigma$ (resp. $\hat{\sigma}$ ) be a semiprime operation on the set of ideals of $A$ (resp. $\mathcal{R}(A, I)$ ). The enough integers $n$ under certain conditions. We first show in this paper, examples of semi-prime operations $\sigma$ and $\hat{\sigma}$ such that $\hat{\sigma}\left[u^n \mathcal{R}(A, I)\right] \cap A=\sigma\left(I^n\right)$ for all integers $n$, and reveal that $A_\sigma(I)=\left\{Q \cap A ; Q \in \operatorname{Ass}_{\mathcal{R}(A, I)}\left(\mathcal{R}(A, I) / \hat{\sigma}\left(u^n \mathcal{R}(A, I)\right)\right)\right\}$ when $n$ is large enough in $\mathbb{Z}$. Finally, we extend these results to filtrations.
Received: December 12, 2023
Revised: February 27, 2024
Accepted: March 15, 2024
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