 Okay, this is Algebraic Geometry lecture nine where we will be discussing the Lasca-Nurter theorem. So for background, let's just recall that Algebraic sets correspond to radical ideals of kx1 up to xn. So this is Algebraic sets of a fine space. So this is just the strong version of the Nore-Stellen sets. We also know that an Algebraic set is equal to a finite union of irreducible Algebraic sets. Now, Algebraic sets correspond to radical ideals. The irreducible Algebraic sets correspond to prime ideals. Just remind everybody that a prime ideal is called prime. If AB in the prime ideal P implies A is in P or B is in P. Another way of putting this is that P is a prime ideal of a ring R if R over P is an integral domain. So it's easy to check that the irreducible Algebraic set is one whose corresponding ideal is prime. So this gives us the following theorem that says any radical ideal P, any radical ideal A is an intersection of a finite number of prime ideals. So this is just the theorem that says an Algebraic set is the union of a finite number of irreducible Algebraic sets. We've translated it into ring theoretic language. And one obvious question is, what about ideals that aren't radical? Can you find some sort of similar decomposition? And the answer is given by Laska, which says that an ideal A of K X1 up to Xn is the intersection of a finite number of primary ideals. So we'll mention what primary ideals are in a moment to theorem. Noto generalized it. So this is also true for notarian rings. So Laska was actually the Laska who was world chess champion for longer than anybody else has been world chess champion. And he actually proved this theorem while he was world chess champion. If you look up his supervisor, his supervisor turns out to be Noto, only it wasn't this Noto. So Laska's supervisor was Max Noto, who was in fact the father of Emmy Noto, who is this Noto who Notarian rings in the Laska Noto theorem is named after. So Max Noto is been somewhat overshadowed by his daughter Emmy Noto, but he was in fact quite a good mathematician in his own right. Anyway, we now need to explain what a primary ideal is. So a primary ideal has two different definitions. First, we have Laska's original definition, which says that P, an ideal P of a ring R is primary. This means that if AB is in P, then A is in P or B to the N is in P for some N. So it's a little bit weaker than the condition that the ideal prime, so this gives the impression that primary ideals are just powers of prime ideals. Well, that's not true in general. Primary ideals and prime ideals are kind of related, but the relations, I mean, in general, primary ideals need not be powers of prime ideals. Well, it turns out to be, it's sometimes better to focus not on the ideal P, but on the module R over P. So R over P has the property that if AB equals naught with A in R and B in R over P, then B equals zero or A to the N equals, and this works for any module over R. So we can actually talk about modules rather than rings, and if it satisfies this condition, we say the module is co-primary. So this condition here for a module is the definition of a module being co-primary. Unfortunately, mathematical notation has got a bit messed up because co-primary is really the important concept, and primary is a sort of more or less a special case of it. So for ideals in the ring, the ideal is primary if and only if the quotient by the ideal is co-primary, but co-primary works for arbitrary modules. Well, it turns out this definition isn't terribly convenient to work with. There's a more convenient definition. So here's an alternative definition. So a module M is co-primary if it has exactly one associated prime. Obviously raises the question, what is an associated prime of a module? Well, an associated prime is a prime ideal so that R over P, so that R over P is isomorphic to a sub-module M. Saying R over P is contained in M is not strictly speaking correct, it's not really a sub-module of this. So this is slightly sloppy notation saying that M contains sub-module isomorphic to R over P, but as usual being precisely correct kind of conflicts with human intuition and it's more convenient to think of R over P as being a sub-module of M and remember that isn't quite correct. So we've got two different definitions. If the ring R is notarian, the two definitions of co-primary are equivalent, which I'm not going to bother to prove but you can find it in any reasonable book on commutative algebra. So this alternative definition turns out to usually easier to use in practice. It's also a bit more natural because it only talks about one module M, whereas the definition of primary ideals talks about two separate ideals. In general, we say that if M contains a sub-module N, then N is called primary if and only if M over N is co-primary. And again, there are two different definitions of that depending on which definition of co-primary you use, but they're equivalent for finitely generated modules over notarian rings. I should say these are equivalent for finitely generated modules, not for arbitrary modules. So with this alternative view of the Laskin-Nurta theorem, the Laskin-Nurta theorem for modules, so these are going to be finitely generated modules over notarian rings, says that zero is an intersection of primary sub-modules of M. Or alternatively, M is actually contained in a finite direct sum of co-primary modules. So originally the Laskin-Nurta theorem was thought to be a theorem about ideals saying that every ideal is an intersection of co-primary ideals. It's really more useful to think of this being a theorem about modules saying that every module is contained in a direct sum of co-primary modules. This should be a finite direct sum. For example, this is pretty similar to a well-known theorem about Abellion groups. So if we look at Z-modules, Abellion groups, then some examples of co-primary modules are the modules Z to the N, where here the prime ideal is just zero or any finite group of order P to the N for some P, and this is a co-primary module whose associated prime ideal is P. So what this says is if you've got a finitely generated Abellion group, it's contained in a direct sum of copies of Z and finite groups of prime power order. Well, if you remember the structure theorem for Abellion groups, you've actually got a slightly stronger theorem. It's actually equal to a direct sum of free Abellion groups and finite groups of order of power of P. Over arbitrary rings, you only get a containment. A module M is not generally equal to a direct sum of this form. So how do you prove Alaska-Nurter theorem? Well, that's originally formidable. Took about a hundred pages, which was one of the longest theorems published at that time. Nurter proved a more general theorem and her proof is very much simpler. In fact, you can give the proof on about a page.