Ratliff-Rush Filtrations associated with ideals and modules over a Noetherian ring
| dc.creator | Puthenpurakal, Tony J. | |
| dc.creator | Zulfeqarr, Fahed | |
| dc.date | 2006-08-20 | |
| dc.date | 2007-01-15 | |
| dc.date.accessioned | 2026-07-07T07:40:35Z | |
| dc.date.available | 2026-07-07T07:40:35Z | |
| dc.description | Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $I$ a proper ideal of $R$. In this paper we introduce and analyze some properties of $r(I, M)=\bigcup_{k\geqslant 1} (I^{k+1}M: I^kM)$, {\it the Ratliff-Rush ideal associated with $I$ and $M$}. When $M= R$ (or more generally when $M$ is projective) then $r(I, M)= \widetilde{I}$, the usual Ratliff-Rush ideal associated with $I$. If $I$ is a regular ideal and $\ann M=0$ we show that $\{r(I^n,M) \}_{n\geqslant 0}$ is a stable $I$-filtration. If $M_{\p}$ is free for all ${\p}\in \spec R\setminus \mspec R,$ then under mild condition on $R$ we show that for a regular ideal $I$, $\ell(r(I,M)/{\widetilde I})$ is finite. Further $r(I,M)=\widetilde I $ if $A^*(I)\cap \mspec R =\emptyset $ (here $A^*(I)$ is the stable value of the sequence $\Ass (R/{I^n})$). Our generalization also helps to better understand the usual Ratliff-Rush filtration. When $I$ is a regular $\m$-primary ideal our techniques yield an easily computable bound for $k$ such that $\widetilde{I^n} = (I^{n+k} \colon I^k)$ for all $n \geqslant 1$. For any ideal $I$ we show that $\widetilde{I^nM}=I^nM+H^0_I(M)\quad\mbox{for all} n\gg 0.$ This yields that $\widetilde {\mathcal R}(I,M)=\bigoplus_{n\geqslant 0} \widetilde {I^nM}$ is Noetherian if and only if $\depth M>0$. Surprisingly if $\dim M=1$ then $\widetilde G_I(M)=\bigoplus_{n\geqslant 0} \widetilde{I^nM}/{\widetilde{I^{n+1}M}}$ is always a Noetherian and a Cohen-Macaulay $G_I(R)$-module. Application to Hilbert coefficients is also discussed. | |
| dc.description | 27 pages. Many minor revisions made, including little changes in title and abstract. Five additional refernces added. To appear in Journal of algebra | |
| dc.identifier | https://arxiv.org/abs/math/0608498 | |
| dc.identifier | http://arxiv.org/abs/math/0608498 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121820 | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13A15(Primary), 13D40, 13A30 (secondary) | |
| dc.title | Ratliff-Rush Filtrations associated with ideals and modules over a Noetherian ring | |
| dc.type | text |