Reductions of monomial ideals

Abstract

In this thesis, we study the normalizations of subrings of a polynomial ring k[x_(1), x_(2),…,x_(n)] where k is a field. We also study the normalizations of monomial ideals of k[x_(1), x_(2),…,x_(n)]. In particular, we find the integral closure I￣ of a monomial ideal I = (x^(n), y^(m)) in k[x, y] and obtain the λ(I￣/I) in terms of n and m. We also construct a saturated chain of ideals from I to I￣ and find the reduction numbers of the ideals in the chain. We also give a method to determine whether an ideal J = (m^(n), f), n ≥ 2, is integrally closed or not when f is a monomial and m = (x, y) is a maximal ideal of k[x, y]. ;다항식환의 부분환과 단항 아이디얼의 normalization을 공부한다. 특히 단항 아이디얼 I=( x^(n), y^(m))의 정폐포(integral closure) I￣를 찾고 I에서 I￣까지 여러가지 아이디얼들의 saturated chain을 구성해서 λ(I￣/I)와 그 아이디얼들의 reduction number 를 직접 계산해본다. 그리고 f가 다항식일때 아이디얼 J=( m^(n), f)가 integrally closed가 될 필요충분조건을 찾았다.