Noetherian Ring - Introduction

Introduction

Let denote the ring of integers; that is, let be the set of integers equipped with its natural operations of addition and multiplication. An ideal in is a subset, I, of that is closed under subtraction (i.e., if, ), and closed under "inside-outside multiplication" (i.e., if r is any integer, not necessarily in I, and i is any element of 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 is a principal ideal ring; that is, for any ideal I in, 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).

Although there are many (equivalent) formulations of what it means for a ring, R, to be Noetherian, one formulation dictates that any ascending chain of ideals in R terminates. That is, if:

is an ascending chain of ideals, then there exists a positive integer n such that

For instance, if I and J are ideals in, there exist integers n and m such that I=(n) and J=(m) (i.e., every integer in I is a multiple of n and every integer in J is a multiple of m). In this case, if and only if every element of I is an element of J, or equivalently, if every multiple of the integer n is a multiple of m. In other words, if and only if m divides n (or m is a factor of n). Furthermore, the inclusion is proper if and only if m is a proper divisor of n (i.e., with k not equal to either 1 or −1).

Thus, if:

is an ascending chain of ideals in and I_j=(n_j) for all j and integers n_j, n_j+1 divides n_j for all j. If each inclusion is proper (that is, if the chain does not terminate), n₂ would be a proper divisor of n₁, n₃ would be a proper divisor of n₂ etc. In particular, which is impossible since there can only be finitely many positive integers strictly less than n₁. Consequently, is a Noetherian ring.

For this reason, the notion of a Noetherian ring generalizes such rings as . The fundamental property of used in the proof above is that there cannot be a chain of positive integers where each integer in the chain is strictly less than its predecessor; in other words, the ring of integers is not "too large" since it cannot sustain such a "large chain". This is typical in the theory of Noetherian rings; often, to prove a result about Noetherian rings, one appeals to the fact that the rings in question are not "too large". More formally, one assumes that the conclusion of the result is false and exhibits an ascending chain that does not terminate thus contradicting the fact that the ring is "not too large", and establishing that the conclusion must, in fact, be true.

While the proof that is a Noetherian ring uses the order structure of, typical proofs in ring theory in general do not assume such additional structure on the ring. In fact, it is possible to give a proof that is a Noetherian ring without appealing to its order structure and this proof applies more generally to principal ideal rings (i.e., rings in which every ideal is generated by a single element).

Although the ring is a Noetherian ring, the theory of Noetherian rings extends far beyond just this ring. For example, let denote the polynomial ring in one indeterminate over . More specifically, let be the set of all polynomials with integer coefficients (such a polynomial is also referred to as a polynomial over ), with addition and multiplication defined to be natural polynomial addition and multiplication. Under these operations becomes a ring. More generally, if R is any ring, the set of all polynomials with coefficients in R can be equipped with the structure of a ring and is denoted by R.

Although every ideal in is simply the set of multiples of a certain integer n, the ideal structure of is slightly more complicated; there are ideals that may not be expressed as the set of multiples of a given polynomial. Put differently, is not a principal ideal ring. However, it is a Noetherian ring. This fact follows from the famous Hilbert's basis theorem named after mathematician David Hilbert; the theorem asserts that if R is any Noetherian ring (such as, for instance, ), R is also a Noetherian ring. By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in , is a Noetherian ring.

Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various "natural rings". While the ideal structure of becomes considerably more complex as n increases, the rings in question still remain Noetherian, and any theorem about that can be proven using only the fact that is Noetherian, can be proven for .