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Integral closures of monomial ideals

Title
Integral closures of monomial ideals
Authors
이은경
Issue Date
1997
Department/Major
대학원 수학과
Keywords
Integral closuresmonomial idealsidealMathematics
Publisher
The Graduate School, Ewha Womans University
Degree
Master
Abstract
Let R = k[x_(1), x_(2), … ,x_(n)] be a polynomial ring over an infinite field k with indeterminates x_(1), x_(2), … ,x_(n). Let N be the set of natural numbers and Γ be a monoid in N^(n). In this thesis, we study definitions and various properties of integral closures of rings and ideals. In particular, we find the explicit form of the integral closure of an ideal generated by monomials in R. We also find the integral closure of a subring of R whose exponents are in Γ. We construct a chain of adjacent ideals in R which exist between an ideal and its integral closure and then find the reduction numbers of those ideals. We showed that m^(k) is integrally closed for all k ≥ 1, where m = (x_(1), x_(2), … ,x_(n)). We also find the reduction number of m^(k) in the case of dimR = 2,3.;x_(l), x_(2), …, x_(n)을 유한체 k상의 미지수라고 할 때 R을 k[x_(1), …, x_(n)] 라 하고, N을 자연수들의 집합, Γ를 N^(n)의 monoid라 하자. 이 논문에서는 ring과 ideal의 integral closure에 대한 정의와 여러 가지 성질들에 대해서 살펴본다. 특히 R상의 monomial ideal의 integral closure의 구체적인 형태를 알아보며, 단항식의 지수가 Γ상에 있는 R의 subring에 대한 integral closure를 살펴본다. 주어진 ideal과 그 ideal의 integral closure사이의 adjacent ideal들의 chain 을 찾아보고 각 ideal들의 reduction number를 구하며, ideal m 이 (x_(l), x_(2), …, x_(n)) 일 때 1보다 큰 정수 k에 대해서 m^(k)가 integrally closed임을 보인다. 또한, dimR이 2,3일 경우에 대하여 m^(k)의 reduction number를 구한다.
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