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On t-Reduction and t-Integral Closure of Ideals in Integral Domains

  • Salah KabbajEmail author
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Let R be an integral domain and I a nonzero ideal of R. An ideal \(J\subseteq I\) is a t-reduction of I if \((JI^{n})_{t}=(I^{n+1})_{t}\) for some \(n\ge 0\). An element x of R is t-integral over I if there is an equation \(x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0\) with \(a_{i}\in (I^{i})_{t}\) for \(i=1,\ldots ,n\). The set of all elements that are t-integral over I is called the t-integral closure of I. This paper surveys recent literature which studies t-reductions and t-integral closure of ideals in arbitrary domains as well as in special contexts such as Prüfer v-multiplication domains, Noetherian domains, and pullback constructions.

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKFUPMDhahranSaudi Arabia

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