On the integral closure of ideals

Among the several types of closures of an ideal I that have been defined and studied in the past decades, the integral closure Ī has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of I are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case I ≠ Ī, √I is still helpful in finding some fresh new elements in Ī\I. Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals.