The Ideal Of The Great Ideal Is Both Covenant

The paper aims to study the questions whether there exists a uniform bound for the reduction numbers of a class of ideals in a Noetherian ring. It is known that even for a local ring R, there exists no such bound for all ideals. In this paper, we restrict our study on the maximal ideals in the quotient rings of a polynomial ring over an infinite field. The main result of the paper states that if R is such a ring, then there is a positive integer n0, such that for every maximal ideal m, there exits elements ci. x2.…, xd€m (d=ht m) satisfying for n≥n0.The main body of this paper is composed of five parts:In the foreword, the background of the research question is introduced;In the second part, we recall some basic knowledge involved in this study, such as some basic concepts, theorems and propositions;The third part discusses the notions of the reduction and closure of an ideal. A rela-tionship between the reduction ideal and the closure of an ideal is given in this chapter. We also recall a basic method of how to construct, an reduction ideal for a m-primary ideal.The fourth part moves to study the power of the maximal graded ideal of a quo-tient ring of polynomial ring over an infinite field. The result of this chapter will play an important role in the proof of the main conclusion of this paper;In the fifth part, we will make use of the results developed in the last parts to prove the main conclusion of this paper, that is. Theorem5.3.