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2017 $t$-reductions and $t$-integral closure of ideals
S. Kabbaj, A. Kadri
Rocky Mountain J. Math. 47(6): 1875-1899 (2017). DOI: 10.1216/RMJ-2017-47-6-1875

Abstract

Let $R$ be an integral domain and $I$ a non\-zero ideal of $R$. An ideal $J\subseteq I$ is a $t$-reduction of~$I$ if $(JI^{n})_{t}=(I^{n+1})_{t}$ for some integer $n\geq 0$. An ele\-ment $x\in R$ 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 investigates the $t$-reductions and $t$-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 corr\-el\-ation with reductions, namely, Section 2 identifies basic properties of $t$-reductions of ideals and features explicit examples discriminating between the notions of reduction~and $t$-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 id\-eals under ring homomorphisms. Throughout the paper, the main results are illustrated with original examples.

Citation

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S. Kabbaj. A. Kadri. "$t$-reductions and $t$-integral closure of ideals." Rocky Mountain J. Math. 47 (6) 1875 - 1899, 2017. https://doi.org/10.1216/RMJ-2017-47-6-1875

Information

Published: 2017
First available in Project Euclid: 21 November 2017

zbMATH: 06816574
MathSciNet: MR3725248
Digital Object Identifier: 10.1216/RMJ-2017-47-6-1875

Subjects:
Primary: 13A15, 13A18, 13C20, 13F05, 13G05

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 6 • 2017
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