Noetherian Rings

Back in my first year, I was at the receiving end of a lot of intellectual 'macho-ism' where quite a few upper-years flexed their ability to waffle by talking about Noetherian rings.

At the time, asides the waffling (and the temporary inferiority complex developed), Noetherian rings sounded so cool! Similarly, even Euclidean rings sounded so cool too. They sounded like some legendary loot you'd find in a sci-fi RPG.

Alas those pie-in-the-sky thoughts were shot down when I learned about both kinds of rings in the 'Groups and Rings' course I took this past semester. They still remain cool though, don't get me wrong; just not as cool as legendary loot you'd find in some sci-fi RPG.

Well, nonetheless, I'll talk about a result regarding Noetherian rings today. But before that, definitions are definitely in order!

(Left/Right) Ideal

Let R be any ring-structure.

A subring, $S \subseteq R$ , is called a left-ideal if $\forall r \in R$ and $\forall i\in S$, we have that $r \cdot i \in S$ as well.

A subring, $S \subseteq R$, is called a right-ideal if $\forall r \in R$ and $\forall i \in S$, we have that $i \cdot r \in S$ as well.

Hence, a subring S for which both definitions hold is called simply as an ideal.

Intuitively, the idea is that an ideal is essentially a 'sticky' pocket-universe within a ring-structure/universe. A left-ideal is 'left'-sticky and a right-ideal is 'right'-sticky.

There's a lot of ideas and results regarding ideal-structures in ring theory, but we'll restrain ourselves for now. In the lead up to the actual result I'm writing this blogpost for, we need to first work with (a couple of) main ideas regarding ideals. So, from here on out, we shall go on a 2-part journey here. The roadmap is this -

We'll first understand how we can add a 'twist' to this idea of an ideal structure from above by 'creating' ideal structures from a single element/member in a ring. Secondly, we'll tie the idea of the good ol' ideal-structure above with the 'synthetic' ideal structures (that we talk about prior to this second step). This fusion of the two is an ideal called a 'principal ideal'. The last milestone in this first half of our journey will thus be talking about the type of overall ring-structure called a Principal Ideal Domain (PID).

The second half of our journey will be in first defining what a Noetherian ring is, and we'll end it off with our main-event - the proof of a result that connects the idea of the PID to the Noetherian ring.

We now look at how we can 'create' an ideal from an element of a ring.

Generated Ideal

Let R be any ring-structure.

$\forall r \in R$, we say that $\langle r \rangle$ is the ideal generated by r, where: $$ \langle r \rangle \coloneqq \{\sum_{i} a_{i}rb_{i} \vert a_{i}, b_{i} \in R\}$$

The way I like to think of a generated ideal is like a pocket-universe which has a beating heart at its center - the 'beating heart' being the ancestor-element (r) from which all other members of this pocket-universe descend from.

Now, what happens if an already-existing ideal-structure $I \subseteq R$ is actually generated, or, equal to an ideal generated by an element? Then, we call this ideal I as a Principal Ideal.

The idea is that any ideal that is principal is nothing but just 'multiples' of some fixed element. A famous example is polynomial rings, where every ideal in those is a principal ideal (multiples of some polynomial). If you want to go even more abstract, then special ring structures called 'Euclidean Domains' exist where every ideal in them is a principal ideal.

Principal Ideal

Let R be any ring-structure.

Take any ideal $I \subseteq R$. Then, we say that I is principal to mean that $\exists r \in R$ where: $$\langle r \rangle = I$$

Finally, I assume familiarity with what an integral domain is, so I will skip ahead and just define what a PID (Principal Ideal Domain) is. If not, this video is a very accessible introduction to integral domains.

PID (Principal Ideal Domain)

We say that an integral-domain structure R is a PID if every ideal $I \subseteq R$ is a principal ideal.

That's the first half of our journey done. We now move to the latter half of our journey by first defining a Noetherian ring and transitioning into the main proof which this article was written for.

Noetherian Ring

Let R be any ring-structure.

We say that R is a Noetherian ring if for any increasing chain of ideals: $$ I_{1} \subseteq I_{2} \subseteq I_{3} \subseteq I_{4}...$$ there exists a cut-off point 'k' such that $\forall$ n $\geq$ k, $$ I_{n} = I_{k} $$

The intuitive idea is that any increasing chain of ideal structures like this stops growing past a certain point and this 'terminal' ideal is the biggest one in this specific chain.

What makes Noetherian rings so cool is that it gives these growing 'sticky' structures a bit of character (asides prime and maximal ideals and the integral-domain/field structure that they endow the quotient ring with, that they create).

Now for the final stretch - the main proof.

THEOREM

Every Principal Ideal Domain is a Noetherian Ring.

Before we go ahead proving this theorem, let's first understand and appreciate the core idea itself. The theorem's basically saying:

'Take any universe where sticky, pocket-universes are each having an atom-like structure: all the elements in each sticky, pocket-universe are bound to a central element. Then, for any increasing nesting of these sticky, pocket-universes; there exists a terminal universe which is the biggest one in this nesting'.

That's pretty cool if you ask me.

Ok, comments aside now, let's prove it!

PROOF:

Suppose we have a PID-structure, denoted by R.

Take any increasing chain of ideals in this universe R: $$ I_{1} \subseteq I_{2} \subseteq I_{3} \subseteq...$$

With this nesting, we consider the union of these nesting-elements/ideals: $$I = \bigcup_{i=1}^{\infty} I_k$$

We first claim that I is also an ideal. This is easy to prove.

PROOF OF CLAIM:

(I omit the proof that I is a subring-structure first because that is a bit mechanical and boring.)

Take any element from R, say, r, and take any element in I, say, i. When we take any element from I, we are picking an element from some $I_{k}$.

So, $$r \cdot i = i \cdot r \in I_{k}$$

Now, since R is a PID, every ideal in R is also a principal ideal. Thus, $\exists d \in R$ such that: $$I = \langle d \rangle$$

Clearly, $$d \in I \implies \exists k \in \mathbb{N}, d \in I_{k}$$

And this second implication means that... $$d \in I_{k+1}, I_{k+2}...$$

Hence, $$(\langle d \rangle \subseteq I_{k}) \wedge (I_{k} \subseteq \langle d \rangle) \implies I_{k} = \langle d \rangle$$

Finally then, for any such nesting, our PID, R, satisfies the definition of a Noetherian ring as well.