Ideais Completos - RIBEIRO, Ranney Ritchie Souto (2023)
The theory of integrally closed ideals in two-dimensional regular local rings (R,m) was introduced by the mathematician Oscar Ascher Zariski. Zariski’s motivation was to give algebraic meaning to the idea of complete linear systems of curves. He studied the class of the contracted ideals. It is known that contracted m-primary ideals I of R are characterized by the following property: (I : m) = (I : x) for some x ∈ m\m2. 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 regular local rings of dimension greater than 2. The m-full ideals are easily seen as full. In this dissertation, we find a sufficient condition for a full ideal to be m-full. We also show that full, m-full, contracted, integrally closed and normal ideals are all equivalents in case of an ideal of parameter. We find a sufficient condition for a basically full parameter ideal to be full.