What are Noetherian Rings and how do they contribute to math.
In: 3
The technical definition is:
> A Noetherian ring is a ring in which there are no infinite (strictly) increasing chains of ideals ordered by inclusion. (This is often stated as “a ring that satisfies the ascending chain condition on ideals”, which means the same thing.)
But my guess is you’ve already encountered that definition. So let’s take an example – I’m going to assume you know what an ideal of a ring is if you’re asking this question at all. [And I’m just going to stop you here, “this isn’t a five year old explanation” people – this five year old is asking about graduate level math, which you’re not going to understand if you don’t have some underlying foundation no matter how it’s explained.]
Consider the integers, as a ring with the usual addition and multiplication. An ideal, over the integers, is just the set of all multiples of some nonnegative integers (so 3Z, the set of all multiples of 3, is an ideal, as is 17Z or 100Z or whatever). This isn’t true for all rings, but it’s one of the “nice” properties a ring can have, and the integers do have it. (More technically: the integers are a principal ideal domain.)
So what would it mean to have an increasing chain of ideals? Well, you’d need one ideal to be contained in the next – that is, you’d need the set of multiples of one number to be a subset of the multiples of another. If you’re at this point in math, it’s probably obvious to you that aZ is a subset of bZ exactly when b divides a, since then every multiple of a is necessarily a multiple of b. So an *ascending* chain of ideals over the integers would correspond to a *descending* chain of positive integers, each of which divides the previous one. But of course such a chain can’t be infinite, since it’s a decreasing sequence of positive natural numbers – which means the original chain of ideals couldn’t be infinite either. So the integers are an example of a Noetherian ring.
An example of a ring that *isn’t* Noetherian (that is, that *does* have infinite increasing chains of ideals) would be the ring of polynomials over infinitely many variables. The ideals generated by (X1), (X1 and X2), (X1, X2, and X3), and so on are definitely strictly increasing (since X2 is certainly not in the ideal generated by X1, and so on), and if you have infinitely many variables you can keep this going forever.
——
The upshot of why we would care about this is that being Noetherian is one constraint on “how big” a ring is. There are a few useful theorems that work for Noetherian rings (by somehow leveraging the finiteness of chains of ideals) but not for non-Noetherian ones.
EDIT: oh, one final note – for noncommutative rings, left and right ideals can behave differently, so there’s a “left Noetherian” and a “right Noetherian”, which don’t necessarily coincide. But if you’re just learning this you’re probably working mostly with commutative rings.
Latest Answers