 This lecture is part of an online course on commutative algebra and will be about artinian rings and modules. So first of all, let's give the definition of an artinian ring. Well, let's first recall the definition of notarian modules. So you remember, there are three conditions that characterize notarian modules. The first is they can satisfy the ascending chain condition for sub modules. Secondly, we can have the condition that every non-empty set of sub modules has a maximal element. And the third condition is that every sub module is finitely generated. And now there's an obvious sort of dual set of conditions. So we call a module artinian if it satisfies the descending chain condition. Well, what's the descending chain condition? The ascending chain condition says that if you've got any ascending chain of sub modules, if these are all strictly increasing, the chain must be finite and the descending chain condition, it's pretty obvious what you have. If you've got a chain of sub modules, then eventually this must stabilize and they must all be the same. The dual of the second condition is obvious. It just says every non-empty set of sub modules has a minimal element. And the dual of condition three, well, I don't know. There doesn't seem to be a good dual of this condition. But anyway, you remember the equivalence of these two conditions just followed from a general property of partially ordered sets. And similarly, these two conditions are equivalent by much the same property of partially ordered sets. So that's an artinian module. So let's see some examples of them. So first of all, let's have some modules that are artinian and notarian. Well, there are quite a lot of these. We can take the zero module over any ring. We can take the z-modulo nz over the integers and more generally any module with a finite number of elements. That obviously has to satisfy the ascending and descending chain condition for sub modules, in fact, for subsets. Another example is any finite dimensional vector space over a field. And we'll see a little bit later that modules that are artinian and notarian are sort of a natural generalization to all rings of finite dimensional vector spaces over a field. Next, we can have examples of modules that are notarian, but not artinian. Well, that's quite easy. For instance, we can take the module z over the ring z. And we saw this as notarian, but it has an infinite descending chain conditions, infinite descending chain of sub modules because z contains 2z, which contains 4z, which contains 8z, and so on. So it's not artinian because you can keep making this smaller and smaller. Another example is we can take the ring z2, which is just the set of all rational numbers A over B with B odd. And again, this is notarian, but not artinian over itself. And we're going to see this a little bit later as the dual of something. And thirdly, we can have modules that are neither artinian nor notarian, such as q over the integers. And here we've got an infinite chain of modules going in both directions because we can take z contains and 2z, contains 4z, and so on. And we can also extend it in the other direction, half z, quarter z, and so on. So we have infinite chain, a chain of modules that's infinitely increasing and infinitely decreasing, so it's not either of them. Finally, we get to modules that are artinian, but not notarian. And these modules tend to be a bit weird. For example, any notarian module, notarian modules tend to be more or less the same as finitely generated modules at least over notarian rings. So these modules tend not to be finitely generated. And we can see an example as we take z a half and then quotient it out by z. So these are all numbers of the form A over 2 to the n. And we can think of this as being isomorphic to a chain of subgroups. So we can take z over z, which is z over 1z. And then this is contained in z over 2z, which we can think of as being a... Well, it's not actually contained in that, but what I mean is it's naturally isomorphic to a submodule of this, 4z. And we can think of that as a submodule of z over 8z and so on. And the union of these is going to be our module m that is artinian, but not notarian. And we can see that these are the only proper submodules that much difficulty. So it's not notarian because there's an infinite increasing chain of submodules, but it's artinian because you can't have an infinite decreasing chain. This turns up as something called the injective envelope of the module z over 2z. And also in some sense it's dual, I can't spell dual, it's dual to the module that was notarian, but not artinian we had earlier. I'm not quite sure exactly what is meant by the word dual here, but if you look at this module over z and this module over z, you can see that they look very much as if they're duals to each other. Well, that's artinian notarian modules and let's do rings. So R is a notarian ring just says that R is a notarian R module. And now it's pretty obvious what the definition of artinian is. R is an artinian ring just means R is an artinian R module. So it just satisfies the descending chain condition for ideals. And in order to get some idea of what these look like, let's just have a look at some examples of artinian and notarian rings. So the ring Z is notarian, but not artinian. And this follows just as we did for modules because it's got an infinite descending chain of sub modules. If we want a ring that's neither notarian nor artinian, well for modules we used Q, but that's no good because Q as a ring is in fact notarian and artinian because it's a field. But if we take something like Z, well this is not artinian, but it's notarian, but we can make it notarian just by adding in polynomials in an infinite number of variables. So this is not notarian or artinian. Informally, being notarian or artinian a condition saying a ring isn't too big in some strange sense of the word big. So if we add lots and lots of variables, we tend to stop a ring from being notarian because this ring is getting too big. Artinian and notarian rings are quite common. There are quite a lot of examples of these. For instance, we can have the integers modulo nz. More generally, we can have any principal ideal domain modulo and ideal. That's not equal to zero. For instance, we can take the ring of polynomials in a field over any modulo, any polynomial. There are lots of non-commutative examples, which are quite important. We won't be covering them in this course, but I'll just briefly list a few. For instance, we can take matrix rings n by n matrices over a field. Or we can take group rings of finite groups over a field, so we take k of g. The reason these are both artinian is that more generally, any algebra over a field that is a finite dimensional vector space over k is artinian and notarian because any ideals must in particular be vector spaces over k. And if you've got a finite dimensional vector space, you can't have infinite increasing or decreasing sequences. There are lots of plenty of commutative examples because we can take, say, a ring of polynomials in two variables and question it out by an ideal of finite co-dimension. And if you remember in an earlier lecture, we showed that there are enormous numbers of ideals of finite co-dimension of the ring of polynomials in two variables. So there are enormous numbers of commutative artinian rings. I mean, there are too many to classify. Finally, we should give an example of a ring that is artinian but not notarian to round off the four possibilities. And I'm not going to do that. And this is not because I'm being obstinate because there actually aren't any examples of such rings. It turns out that all artinian rings are notarian, which is, it's actually pretty surprising, at least to me, because artinian and notarian seem like dual properties. But for rings, they're not really dual because one of them actually implies the other. This actually wasn't known in the early days of when people were studying artinian rings. So here is, it's actually Artin's book on artinian rings. He didn't call them artinian rings. He called them rings with minimum condition. They were only named after him later. And you see, he sort of says here, it was formally found expedient to impose a maximum as well as a minimum condition on ideals. That means the maximum condition says that the ring is notarian and the minimum condition says it's artinian. So he says that formally people would talk about rings that were artinian and notarian. And it was only later discovered that the artinian condition actually implies the notarian condition. So that's artinian rings. We're now going to... So next lecture, we'll be proving that artinian rings are automatically notarian and classifying them. But in preparation for that, we want to study the modules that are artinian and notarian. So first of all, we say a module is simple if the only sub-modules, not an M, and M is not equal to zero. The zero module is not usually countered as simple for the same reason that one isn't countered as a prime. So for example, Z over PZ is a simple Z module and K is a simple K module. So more generally, the simple modules over R are just the modules isomorph to R over M for M maximal. This is very easy to prove, so I'll just leave it as a 30-second exercise. And more generally, we can build modules over simple modules. So M is called module of finite length and if we can find a chain of sub-modules, nought equals M nought contain an M1, and so on up to Mn equals M. So Mi over Mi minus one is simple. In other words, M is built out of a finite number of simple modules and the length of this chain, N, is called the length of M. Well, you may be a little bit worried about that because there can be several different chains of sub-modules of M and it's not immediately clear that they all have the same length and that's something we're going to be proving fairly shortly. Anyway, let's just give some examples of modules of finite length. So for example, a finite dimensional vector space is obviously finite length and we should think of the finite length modules as being sort of analogs of finite dimensional vector spaces. They're about the same size in some rather vague sense of the word size. A second example is just the group Z over P to the NZ. So, you know, you can build up a chain of sub-modules, nought contain in Z over PZ, contain in Z over P squared Z. Well, it's not really contained in what I mean as there's an isomorphism from this to a sub-module but whatever and so on up to Z over P to the NZ and the quotients are all Z modulo PZ. So, you notice this is not a direct sum of simple modules. The simple modules are sort of joined together in a somewhat more complicated way. So, what we'll now show is that M has finite length is equivalent to M being notarian and artinian. So, in one direction, so if you want to prove that finite length implies notarian and artinian, we first of all notice that simple obviously implies notarian and artinian. And next we notice that if Z modulo PZ implies artinian and next we notice that if we have an exact sequence nought goes to A, goes to B, goes to C, goes to nought. So, if this is exact, then A and C notarian implies B is notarian, which we may have proved earlier and A and C artinian implies B is artinian. And you get this just by taking the proof of things and kind of turning it upside down. Anyway, these statements are both very, very easy, so I'm not going to bother proving them. So, we see from this that if A and C have finite length, then B also has finite length. So, A, C finite length implies that B has finite length. So, anyway, so this implies that if M has finite length, then A and C are both notarian. So, conversely, if M is notarian and artinian, let's prove it as finite length. What we do is we start with a zero module of M. Then we take M1 to be a minimal non-zero, unless the module is zero, in which case we're done. And we can do that because the module is artinian. So, among the non-zero modules, we can choose a minimal one. If M1 isn't equal to M, then we choose a minimal module that's strictly bigger than M1. And again, we can do this because the ring is artinian. So, we keep going like this. Now, we've got a chain of strictly increasing modules, and this must actually stop. So, this stops as M is notarian. So, a module has finite length, if and only if it's both notarian and artinian. So, the last theorem we're going to have just says that if M has finite length, any two chains, let's say any two maximal chains have the same length. So, that's chains you can't increase by squeezing in an extra module by two of the modules. So, if you've got two chains, nought equals M nought contained in M1, contained in M. And the second chain, nought contained in M1 and so on up to n. Then, we form a sort of square grid like this. And we put modules in all entries of this grid. And the modules we're going to put in here are going to be things of the form. All these modules will be things like I, intersection, M, J, N, J. So, the I, J's position will be the intersection of these two modules. So, we've got a sort of rectangular array of inclusions of modules. And now, let's look at each square. So, in each square, we've got four modules here. And we look at what the quotients are. Well, the quotient of this module by this module must either be 0 or it's the quotient of the corresponding module, n's, as you can see. So, this quotient is either simple or 0. And this is either simple or 0. And this is either simple or 0. And what you can see is the only possibilities are that either this quotient here is equal to that quotient there. And this quotient here is equal to that quotient there. Or these two modules are the same in these two modules are the same so the quotients are the same and we've got two quotients that are the same there. So in each square we either have these two quotients the same and these two quotients are the same or these two quotients are the same and these two quotients are the same. And now this immediately implies that any two chains have the same length and in particular must have the same quotients in them because you remember one chain we sort of get by going along here and taking successive quotients and each of the quotients is either going to be zero or one of the simple modules in this chain. And the other quotient was got by going along this route here so you can think of this as being a sort of taxi cab one and round from a square grid. And now we can turn this taxi cab route into the other one by changing it one square at a time for instance we can change to this route here and then we can change to say this route here and then we might change to say this route here and each time you're changing the modules in your chain just by switching from one side of a square to the other side of a square. And if you look at the only possibilities for the square this may change the order of the simple modules but it doesn't change which simple modules you have or how many of each sort of simple module. So for any two chains with simple quotients the number of times any given simple module occurs is the same. So let's just write down the obvious consequences of it. So first of all the length of a simple module of a finite length module is well defined. And secondly the length is additive on exact sequences. In other words if we've got the sequence naught goes to a goes to b goes to c goes to naught and abc a finite length then the length of a plus the length of c is equal to the length of b and that follows because if you've got a sort of series of sub modules of a with simple quotients and the series of sub modules of c with simple quotients you can just splice those sequences together and get a series of sub modules of b. Notice that the length is sort of analogous to the dimension of a vector space. In fact it is exactly the same as the dimension of a vector space from modules over from modules over vector spaces. This theorem is very similar to the Jordan Holder theorem in finite groups which says that if you split up a finite group into simple finite groups then the number of times each finite group finite simple group occurs is independent of how you split them up. In fact they're both special cases of a more general theorem about groups with operators on them where you can think of a module over a ring as being an abelian group with with the elements of the ring acting on it as operators. Okay so next lecture we will be studying the structure of artinian rings and showing that they're all notarian and showing that they're products of local artinian rings.