 At this moment in our lecture series, this is the appropriate time to introduce the idea of a noetherian ring named after, of course, Emmy Nerder, a very famous algebraist from the 20th century. We say that a ring is noetherian if it satisfies the so-called ascending chain condition, often called ACC for short, the ascending chain condition on its ideals. So what that means is if we have an ascending chain of ideals, we have some ideals, E1, E2, E3, et cetera, and then it's an ascending chain. So E1 contains E, excuse me, E2 contains E1, E3 contains E2, E4 contains E3, et cetera, et cetera, et cetera. The ascending chain condition says that whenever we have an ascending chain of ideals, there exists some integer in such that every ideal in the ascending chain is actually equal to I in when you're larger than that. That's to say that the ascending chain condition tells us that eventually every ascending chain of ideals stabilizes. It reaches some point and then it's fixed at that ideal forever afterwards, okay? The idea of a noetherian ring is closely connected to the idea of an artinian ring named after Emile artin, of course. An artinian ring is a ring that satisfies the descending chain condition or DCC for short, which it's defined analogously that if we have a bunch of ideals, I1, I2, I3 all the way up to infinity, right? Now if we have a descending chain, so I1 contains I2, which contains I3, which contains I4, then the descending chain condition tells us that eventually there's gonna be a number N for which once you hit capital N, every ideal after that point is actually equal to I in. So it stabilizes. And so the descending chain condition, just as again a summary says that every descending chain of ideals stabilizes at some point. Now these two notions, of course, sound very similar. The ascending chain condition, the descending chain condition, and in many algebraic categories, these two notions are independent of each other. You can have a noetherian algebra that's not artinian and you can have artinian algebras that are not northerian. Module theory, of course, is a place where such a thing plays out. But when it comes to commutative rings, again, I'm gonna just state this without proof, every artinian commutative ring is in fact noetherian. That for commutative rings, if you have the descending chain condition, that implies the ascending chain condition for which the proof of that's gonna go beyond the scope of our lecture series. So I mostly just wanted to mention this for the sake of exposure. The reason we introduce this is really we wanna talk about noetherian rings, which are gonna be important with our conversation of factorization in integral domains. So our reason for noetherian rings is actually the following theorem right here. Every principal ideal domain is in fact a noetherian ring, a noetherian domain here. That is the principal ideal domain satisfy the ascending chain condition. So to prove that, we're gonna start off with an arbitrary ascending chain of ideals inside of our principal ideal domain D, like so. And so the bulk of the argument's gonna come from the following claim here. If we have an ascending chain of ideals, if I take the union of that entire chain, that itself is an ideal inside of the ring. This is a very important result and it's not, I don't even need, I don't even need the principal ideal hypothesis to make this. This is actually just true for arbitrary rings. Every ascending chain, excuse me, if you take the union of any ascending chain of ideals, that itself will be an ideal. Similarly, you can also take the intersection of a descending chain of ideals, that'll also be ideals, but intersections of ideals are always ideal. So that one's not very surprising. Unions in general are not, unions of ideals in general are not ideals, but if we take the union of an ascending chain, that is always an ideal. So what do we have to show? To show that this is an ideal. We'd have to show something like it contains the zero element, right? Well, each and every one of the ideals in the chain contains the zero element. So in particular, I1 has it, so the union will have it. So it has the zero element. Is it closed under addition? Okay, well, suppose we have two elements, A and B, which belong to the ideal. Well, A has to show up somewhere, right? If A is inside of I, that means one of the ideals, we'll call it I sub I, contained A. Actually, excuse me, we'll call it I sub N in this conversation. But then the element B also has to be contained inside of one of the ideals. We'll say B is inside of IM. Now, because this is an ascending chain, it means either IN contains IM or IM contains IN. One of them contains the other. So in other words, you can just look at the index, right? Because as the index gets bigger, the ideal gets bigger. And so without the loss of generality, we can assume that M is less than or equal to N. And so in particular, that tells us that IM is a sub-ideal of IN. It's a subset of IN. So that both the elements A and B belong to the ideal IN. As it's an ideal, it's closed under addition, okay? So we can add A and B and it'll be inside the ideal, like I said, zeros inside of there. And then by similar reasoning, right? If we take this element A and I times it by anything in the domain, the product AR will be inside of IN because it's an ideal and that's a subset of I. So the sum, the zero, this product, this ideal closure, all of these things are contained inside of I because at some finite step along the way, it was an ideal and that ideal had this property. So the union of an ascending chain is always gonna be an ideal. So if you have this ascending chain of ideals, its union is an ideal. That's a very important property. Again, that stands alone by itself. I didn't need to have any hypothesis about principal ideals or domains or anything like that. But the fact that we do have a principal ideal domain is exactly gonna come into the forefront right now. The union of your ascending chain, I, is an ideal and therefore it's a principal ideal inside of a PID. So there has to be some element X which generates the union of this ascending chain because it's an ideal. So the union is just gonna be the principal ideal generated by X. But if that means in particular the element X is inside of this ideal, this ideal I is the union of ideals. So X only got inside that union because one of the ideals, at least one of the ideals contained X. So we'll say that the index of an ideal, we'll say the first ideal that contained X call that capital N because X is belonging to that ideal I-N right there. So since X belongs to the ideal I-N, that means it contains the entire principal ideal generated by X, all right? And as we go up the ascending chain, every other element, every other ideal in the chain that's after capital N will have to also contain the principal ideal generated by X because it contains X because X was in I capital N. And so if we take the union of all these things, clearly every later ideal in the chain is contained in the union, but the union is itself equal to the principal ideal generated by X. And so we have this chain of inequalities that start and end with the same set. So this actually forces there to be equality on everything after that point. So once we hit the marker capital N in our ascending chain, everything stabilizes to be the principal ideal generated by X. And therefore this shows you that principal ideal domains are in fact, noetherian. In fact, you can modify the proof that we just saw right here to show that every ring with the property that all ideals are finitely generated such a ring is going to be noetherian. I'm gonna leave that of course as an exercise for the viewer here that finite generation is closely connected to this idea of ascending chain conditions. All right, so by the theorem we've just proven here, all principal ideal domains are noetherian. And so in fact, the integers and the Gaussian integers, these are principal ideal domains. So these are examples of noetherian rings. Now, we've often looked at the ring, Z join square of negative three as a counter example to these factorization principles we're learning about here. Like Z join square of negative three was not a unique factorization domain. It's not a principal ideal domain, but it is still actually noetherian. It's a noetherian domain. It takes a lot of effort to come up with a integral domain that is not noetherian. And the proof that Z join square of negative three is noetherian, I'm gonna leave that to the viewer. Mimicking techniques we've set already in this lecture series, you could show that it satisfies the ascending chain condition. You could also prove that every ideal in Z joined square of negative three is finitely generated. Like I said, that's equivalent to the ascending chain condition, although I'm leaving that as an exercise to the viewer to prove that one. And so in fact, pretty much every ring we have studied in this lecture series is in fact noetherian. Even infinite algebra is like we've seen right here, right? This is an infinite set because I keep on coming back to this idea of finite generation. Clearly, if you have a finite ring, it's finitely generated because the entire set is finite. But a noetherian ring will be one where every ideal is finitely generated. And that happens even for Z to join the square of negative three. And this is a common feature we see in mathematics that even when we are in an infinite setting, we often try to grasp something that's finite. Take for example, linear algebra. You take the idea of a vector space. Take something like Rn, right? As a set, Rn is an infinite set. It's uncountable. It'll have the same cardinality as the real numbers themselves, the continuum. But in linear algebra, we're very fixated on finite dimension, infinite dimensional linear algebra often called functional analysis. It's very, very different than the typical linear algebra you see with finite dimensional vector spaces in like a class like math 2270 at SUU. And this idea of noetherian or artinian is another idea to try to put a finite condition on potentially infinite rings. Like I mentioned, the ascending chain condition is equivalent to the ideals being finitely generated. The descending chain condition is also closely related to that. We saw for commutative rings, the descending, well, we mentioned it, we didn't prove it, but the descending chain condition implies the ascending chain condition. So these are all ways of putting a finite label on things, okay? But even in the case of like, in functional analysis, when you have infinite dimensional vector spaces, you didn't move from your Hamel basis to like a Hilbert basis. And then you talk about finite dimensional Hilbert basis and things like that. We're always trying to grasp to something finite. So the skinny of this little narrative right now is that a non-noetherian ring is truly infinite. It's more infinite than even these integral domains right here. And so such a ring to describe would have to be, let's give me a lot more complicated than we've seen with. So we can rest assured that in our lecture series, we won't really talk much about non-noetherian rings and we're gonna be okay with that.