t-reductions and t-integral closure of ideals

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3 Scopus citations


Let R be an integral domain and I a nonzero ideal of R. An ideal J ⊇ I is a t-reduction of I if (JIn)t = (In+1)t for some integer n ≥ 0. An element x ϵ R is t-integral over I if there is an equation xn + aixn-1 + ⋯ + an-1x + an =0 with ai ϵ (Ii)t for i = 1, ., n. The set of all elements that are i-integral over I is called the i-integral closure of I. This paper investigates the t-reductions and i-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well known results in the literature, on the integral closure of ideals and its correlation with reductions, namely, Section 2 identifies basic properties of t-reductions of ideals and features explicit examples discriminating between the notions of reduction and i-reduction. Section 3 investigates the concept of t-integral closure of ideals, including its correlation with t-reductions. Section 4 studies the persistence and contraction of t-integral closure of ideals under ring homomorphisms. Throughout the paper, the main results are illustrated with original examples.

Original languageEnglish
Pages (from-to)1875-1899
Number of pages25
JournalRocky Mountain Journal of Mathematics
Issue number6
StatePublished - 2017

Bibliographical note

Funding Information:
This work was supported by King Fahd University of Petroleum & Minerals, DSR grant No. RG1328.

Publisher Copyright:
Copyright © 2017 Rocky Mountain Mathematics Consortium.


  • Basic ideal
  • Integral closure of an ideal
  • Prüfer domain
  • PvMD
  • Reduction of an ideal
  • T-ideal
  • T-integral dependence
  • T-invertibility
  • T-operation
  • T-reduction

ASJC Scopus subject areas

  • Mathematics (all)


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