Integrally Closed Ideals and Some Examples

Abstract

Let (R, m) be a 2-dimensional regular local ring with the residue field k. An integrally closed ideal is a contracted ideal but the converse is not true in general. In this thesis we review various properties of contracted ideals and those of integrally closed ideals. We also find various examples which illustrate various properties of integrally closed ideals and contracted ideals. In particular, we construct some examples of contracted ideals that are not integrally closed, and then find the integral closures of those ideals. Transforms of integrally closed ideals are also integrally closed ideals. We study properties of the transforms of integrally closed ideals and we find some related examples.;(R, m)을 2차원적 정규국소환이라 하자. 일반적으로 integrally closed ideal을 contracted ideal이지만 그 역은 성립하지 않는다. 이 논문에서 우리는 contracted ideal들과 integrally closed ideal들의 다양한 성질들을 살펴보고 그 성질들을 설명해주는 여러 가지 예들을 찾는다. 그리고 integrally closed idea의 변환이 integrally closed ideal이기 때문에 integrally closed ideal의 변환에 관한 성질을 공부하고 그것에 관계되는 예들을 찾는다.