Bounds on the Ratliff-Rush Index and the Castelnuovo-Mumford Regularity

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\ge 2,$ and $I$ an $\mathfrak{m}$-primary ideal of $R\mathrm{.}$ Denote $r_J(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $\rho (I)$ as the Ratliff-Rush index of $I$. We establish upper bounds on $\rho (I)$ in terms of Hilbert coefficients $e_i(I)$ for $0\le i\le d+1,$ and $r_J(I)\mathrm{.}$ Suppose $\widetilde{I^{r_J(I)}}\mathrm{\ne }{I}^{r_J(I)}\mathrm{.}$ We prove that $\rho (I)\le r_J(I)-1+(-1{)}^{d+1}(e_{d+1}(I)-{\tilde{e}}_{d+1}(I))\mathrm{.}$ When $d=2,$ we prove that $\rho (I)\le r_J(I)-1+(e_2(I)-1)e_2(I)-e_3(I)\mathrm{.}$ This established bound on $\rho (I)$ consequently leads to a bound on the Castelnuovo-Mumford regularity of the associated graded ring of $I\mathrm{.}$ We also determine bound on $\rho (I)$ in two-dimensional Buchsbaum rings with positive depth.