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INTEGRAL AND COMPLETE INTEGRAL CLOSURES OF IDEALS IN INTEGRAL DOMAINS

A. MIMOUNI

Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, P. O. Box 278, Dhahran 31261, Saudi Arabia

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https://doi.org/10.1142/S0219498811004884Cited by:1 (Source: Crossref)

Abstract

This paper studies the integral and complete integral closures of an ideal in an integral domain. By definition, the integral closure of an ideal I of a domain R is the ideal given by I′ ≔ {x ∈ R | x satisfies an equation of the form x^r + a₁x^r-1 + ⋯ + a_r = 0, where a_i ∈ Iⁱfor each i ∈ {1, …, r}}, and the complete integral closure of I is the ideal Ī ≔ {x ∈ R | there exists 0 ≠ = c ∈ R such that cxⁿ ∈ Iⁿfor all n ≥ 1}. An ideal I is said to be integrally closed or complete (respectively, completely integrally closed) if I = I′ (respectively, I = Ī). We investigate the integral and complete integral closures of ideals in many different classes of integral domains and we give a new characterization of almost Dedekind domains via the complete integral closure of ideals.

This work was supported by KFUPM.

Keywords:

AMSC: Primary: 13A15, Primary: 13A18, Primary: 13F05, Secondary: 13G05, Secondary: 13F30

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