Among the several types of closures of an ideal I that have been defined and studied in the past decades, the integral closure \(\) 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 \(\), \(\) is still helpful in finding some fresh new elements in \(\). Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals.
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