Noetherian ring

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In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:

there exists a positive integer n such that:

There are other equivalent formulations of the definition of a Noetherian ring and these are outlined later in the article.

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

Contents

Introduction

Let \mathbb{Z} denote the ring of integers; that is, let \mathbb{Z} be the set of integers equipped with its natural operations of addition and multiplication. An ideal in \mathbb{Z} is a subset, I, of \mathbb{Z} that is closed under subtraction (i.e., if a,b\in I, a-b\in I), and closed under "inside-outside multiplication" (i.e., if r is any integer, not necessarily in I, and i is any element of I, ri=ir\in I). In fact, in the general case of a ring, these two requirements define the notion of an ideal in a ring. It is a fact that the ring \mathbb{Z} is a principal ideal ring; that is, for any ideal I in \mathbb{Z}, there exists an integer n in I such that every element of I is a multiple of n. Conversely, the set of all multiples of an arbitrary integer n is necessarily an ideal, and is usually denoted by (n).

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