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Ratliff-Rush Monomial Ideals

SPEAKER: Veronica Crispin

TITLE: Ratliff-Rush Monomial Ideals

ABSTRACT: In a paper from 1978 Ratliff and Rush studied the union $tilde{I}=bigcup_{lgeq 1}^{infty} (I^{l+1}:I^l)$. They show that $(tilde I)^l=I^l$ for sufficiently large $l$ and that $tilde I$ is the largest ideal with this property. We call $tilde I$ the Ratliff-Rush ideal associated to $I$, and an ideal such that $tilde I=I$ a Ratliff-Rush ideal. The operation $:tilde{}:$ cannot be considered as a closure operation in the usual sense, since $Jsubseteq I$ does not generally imply $tilde Jsubseteqtilde I$. One of the reasons to study Ratliff-Rush ideals is that for a regular $m$-primary ideal $I$ in a local ring $(R,{m,k})$ the ideal $tilde I$ can be defined as the unique largest ideal containing $I$ and having the same Hilbert polynomial as $I$. We show how to compute the Ratliff-Rush ideal associated to a monomial ideal in a certain class in the ring $k[x,y]$ ($k[[x,y]]$) and find an upper bound for the Ratliff-Rash reduction number for such an ideal. We start by giving some results about numerical semigroups that are crucial for our later work. We conclude by duscussing several examples.