Primary decomposition of monomial ideals

Abstract

Let k be a field and R be a polynomial ring in n variables over k. Every ideal I of R=k[x₁, ... ,x_n]can be written as a finite intersection of primary ideals. A monomial ideal is an ideal of k[x₁, ... ,xn] that can be generated by monomials in the variables x₁, ... ,x_n. In this thesis, we study definitions and various properties of primary decompositions of ideals in k[x₁, ... ,x_n]. In particular, we study various properties of primary decompositions of monomial ideals in k[x₁, ... ,x_n]. We find a minimal primary decomposition of 0 in k[x₁, x₂]/I for a monomial ideal I in k[x₁, x₂] and in k[x₁, x₂, x₃]/J for a monomial ideal in k[x₁, x₂, x₃]/J.;이 논문에서는 monomial ideal의 primary decomposition을 구하는 방법을 찾아보았고, 0의 minimal primary decompoisition이 unique하지 않는 예를 variable이 2개일 때와 3개일 때 각각 찾아보았다.