Some Properties Of Ideal Power

Let k be a field and k[X₁, X₂] be the polynomial rings in two variables. For a polynomial F（X₁, X₂）, let us consider the ring R=k[X₁, X₂]/（F（X₁, X₂））. The paper aims to study the following question whether there exists an integer t such that for any ideal I of R, there an element g of I satisfying for n≥t.We will prove that the integer t may be chosen as the degree of the polynomial of F（X₁, X₂） in several cases.The main body of this paper is composed of three parts.In the foreword, we present the the background of the question.In the second part, we recall some basic knowledge involved in this study, such as some basic concepts and theorems.The third part is the main content of this paper. Because we could not find a unified way to discuss the problem, we proceed with this part according to the type of F（X₁, X₂）. In the section 3.1, we discuss F（X₁ X₂）= X₁t. In section 3.2, we consider the case F（X₁,X₂）= F（X₁）. The third section discusses the case of F（X₁, X₂）=X₁X₂。In this section, we find that the ideals of R are divided into two classes. One of them are principal ideals, and the ideals in the other class are also satisfied with the conclusion. In the fourth section we discuss F（X₂,X₁）=（X₁X₂）2, and it turns out that the conclusion is also valid in this case.