INTEGRAL AND COMPLETE INTEGRAL CLOSURES OF IDEALS IN INTEGRAL DOMAINS

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 xr + a1xr-1 + ⋯ + ar = 0, where ai ∈ Ii for 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 ∈ In for 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.