A Less Restrictive Brian\c{c}on-Skoda Theorem with Coefficients

• Main Authors: Aberbach, Ian M, Hosry, Aline
• Format: Journal Article
• Language: English
• Published: 06.10.2010
• Subjects: Mathematics - Commutative Algebra
• DOI: 10.48550/arxiv.1010.1062

Journal of Algebra, 345 (2011), 72-80 The Brian\c{c}on-Skoda theorem in its many versions has been studied by algebraists for several decades. In this paper, under some assumptions on an F-rational local ring $(R,\m)$, and an ideal $I$ of $R$ of analytic spread $\ell$ and height $g < \ell$, we improve on two theorems by Aberbach and Huneke. Let $J$ be a reduction of $I$. We first give results on when the integral closure of $I^\ell$ is contained in the product $J I_{\ell-1}$, where $I_{\ell-1}$ is the intersection of the primary components of $I$ of height $\leq \ell-1$. In the case that $R$ is also Gorenstein, we give results on when the integral closure of $I^{\ell-1}$ is contained in $J$.