Full Ideals

Contractedness of m-primary integrally closed ideals played a central role in the development of Zariski's theory of integrally closed ideals in two-dimensional regular local rings (R, m). In such rings, the contracted m-primary ideals are known to be characterized by the property that I : m = I : x for some x is an element of m\m(2). We call the ideals with this property full ideals and compare this class of ideals with the classes of m-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The m-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be m-full. We also show the equivalence of the properties full, m-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full.