 This lecture is part of an online Commutative Algebra course. So this is the introductory lecture where I'm just going to give some background reading and give a very brief overview of the course. So the level will be roughly that of a first year graduate course. So I assume anybody in it actually knows what a Commutative Ring is. So Commutative Algebra is a sort of service course for various other courses. So you can think of Commutative Algebra as sort of sitting in the middle of various other courses. And then we might have here is Algebraic Geometry and another course that uses a lot of Commutative Algebra is Number Theory. And the third one that used to use a lot of Commutative Algebra which is kind of a bit out of fashion these days is Invariant Theory. So a lot of the original work on Commutative Algebra was actually motivated by problems in Invariant Theory. And the philosophy of this course is that Commutative Algebra is really something you use in Algebraic Geometry and Number Theory rather than something you study for its own sake. So this is directed mainly towards people who want to go and do research in Algebraic Geometry or Number Theory and need Commutative Algebra in order to do so. So let's start by just giving some of the basic examples of Commutative Rings that we study. So in Number Theory, we have the following fairly typical examples of Commutative Rings. First of all, we have the integers Z. Then we have things like rings of integers in Algebraic Number Fields. For instance, you might take the ring Z i of all numbers n plus n i with i squared equals minus one. So this is the Gaussian integers. Or you might take a cyclotomic field. You might take all the numbers generated by zeta where zeta might be say e to the two pi i over 23. So zeta to the 23 equals one. And this would be a cyclotomic field in Number Theory. And a typical sort of question that Commutative Algebra might ask about these rings is, is this ring a unique factorization domain? For example, can every element in it be uniquely factored as a product of primes, opto, order and units and so on? And for instance, these two are unique factorization domains. And this one, as Kummer showed, is not a unique factorization domain. And Commutative Algebra produces invariants like the Picard group, pic of R of a ring, which consists of roughly the invertible modules that we will discuss later and kind of measures how far the ring is from having unique factorization. Next, we look at some examples from Algebraic Geometry. So a typical example of a commutative ring in Algebraic Geometry is a coordinate ring of an algebraic variety. This just means a ring generated by the coordinate functions. And the simplest example would be something like a ring of polynomials in two variables over a field. So X and Y are the coordinates of the plane. So this is just the ring generated by coordinates. And of course, you can add more variables if you like. So in the early days of Commutative Algebra, this was really the only sort of ring people really considered, well, that they would also consider polynomials over the integers. But Commutative Algebra basically meant for theory of polynomials over a field. And you can have slightly more complicated examples. For example, you might take an elliptic curve, Y squared equals X cubed minus X. So this looks something like this. Might have an elliptic curve. And you can look at the coordinate ring of this elliptic curve, which means roughly speaking, the functions on the elliptic curve that are equal to polynomials. And this ring is just a quotient of the coordinate ring by the ideal generated by Y squared minus X cubed plus X. So the ideal generated by some elements is normally indicated by putting it in parentheses. So this would be a fairly typical example of a ring that might be of interest to an algebraic geometry. And you can ask various questions about this. For instance, you could be very lazy and just ask the same question we had last time. Does this ring have unique factorization? And in fact, this ring doesn't because it has something called a Picard variety, which is quite big. And you can ask other questions like, how are points of this curve related to the ring R? In other words, suppose you start with an elliptic curve and form its coordinate ring, and then you very carelessly lose the elliptic curve and can't remember what it is, but all you have is the coordinate ring. And the question is, can you reconstruct the elliptic curve from the coordinate ring? And in fact, you can, for instance, a point of the elliptic curve corresponds to a homomorphism from the ring R to the field K, because you can just think of a point as corresponding to the function taking the value of a coordinate function at that point. So points kind of correspond to homomorphisms from R to K. And if you've got a homomorphism from R to K, this corresponds to an ideal of R. This correspondence isn't one to one, but points of the elliptic curve turn out to be related to ideals of the ring R. So one of the things commutative algebra has to do is explain the relation between ideals of the coordinate ring and the geometry of the algebraic curve. Next, we have some examples from invariant theory. So what is invariant theory? Well, invariant theory looks like this. Suppose you take, say, some sort of platonic solid. You might take, say, an icosahedron looking like this, sitting in three-dimensional space. So we have an icosahedron in R3. And you might look at the symmetries of the icosahedron and the symmetries of the icosahedron form a group of order 120 because it's got 20 faces. And once you fix a face, there are three possible rotations and also some reflection. So you get 120 symmetries. And the group of symmetries acts on R3. So on the ring of polynomials on R3, so it acts on the ring Rxyz. And now we can ask the following question. What are the invariants? These are just the polynomials fixed by the symmetries. In other words, we want a polynomial function on R3, such that if you rotate R3 preserving this icosahedron, then that polynomial remains the same. Well, there's an obvious invariant, which is just x squared plus y squared plus z squared, assuming you put the origin at the center of the icosahedron. And if you put the origin anywhere else, you'll be being pretty stupid. So this is an invariant because it's basically just the distance or distance squared or something. And you can ask, what are the other invariants? Well, the invariants obviously form a ring called the invariant ring, because the sum and product of two invariants is obviously invariant. And you can ask, what is the structure of this ring? And Kleiner showed that the invariant ring is a polynomial ring in three generators, A, B and C, where A is this generator here, which has degree two, and B has degree six, and C has degree 10. So one particularly important question is, is the ring of invariants finitely generated? It means, can you find a finite number of invariants such that any other invariant is a polynomial than those? And this was a really fundamental problem in invariant theory back in the 19th century. And the answer turns out to be sometimes it is finitely generated and sometimes it isn't. And this is one of the things we will be discussing. So in particular, we will be proving Hilbert's theorem, which says that quite often the invariant ring is finitely generated. So next I'll discuss some background reading for the book. First of all, the course is going to be mainly following the book by Eisenberg on commutative algebra with a view towards algebraic geometry. In particular, most of the topics I will be talking about are somewhere in this book. And in the description below the video, I might try and remember to list some background reading from this book and also some exercises in case you want exercises. A quite similar book is the classic book, Introduction to Commutative Algebra by Atea and McDonald. This is at a similar level to Eisenberg's book. The only difference is it contains rather less. As you can see, it's only about a quarter of the thickness. So in particular, Atea and McDonald doesn't actually contain all the topics we're going to be covering. Another classic book at a very similar level is the one by Zariski and Samuel Volume I, Commutative Algebra, which is an old classic. It's recently been reprinted as a low-cost paperback by Dover. For some more advanced books, I'll just briefly mention them. So the next four books will discuss more advanced topics, many of which we won't be covering in the course. First of all, there's Volume II of Zariski and Samuel, which covers things like valuation rings and technical theorems about power series rings in much more detail. There's a book by Seher, which like absolutely every book by Seher is incredibly clearly written and well worth reading. I mean, you should, if you see a book by Seher, you should buy it and read it no matter what the topic is. Then there's an old book by Nagata on local rings. So, let's see if I can find the, here we are, local rings by Nagata. This again contains mostly somewhat more advanced topics and is particularly notorious because it's got this appendix at the end, which contains various examples of awful rings. Nagata explicitly calls them bad rings. So I'm not just making this up, where he gave counter examples to all sorts of things that people were sort of hoped might be true for commutative rings. And unfortunately, commutative rings turned out to be worse than people thought. Then there's the book Commutative Algebra by Matsumura, which is one of the books used by Robin Hart, Sean in his Algebraic Geometry book as a reference for commutative algebra. That's one of the few books that actually contains a definition of an excellent ring. And we're not going to be covering excellent rings in this course because I can never actually remember the definition of an excellent ring. So, next we move on to a couple of encyclopedic volumes. The first one is Borbaki's book on commutative algebra, which used to be seven chapters. They've recently started adding further chapters to it. So we've now got up to chapter 10, as long as you read French. I don't think the English translation for chapters eight to 10 has come out yet. Borbaki's book is a sort of a reference work. It contains almost everything you would need to know it in commutative algebra in the greatest possible generality, but it may not be the easiest thing to use as a textbook. Another encyclopedic source of information on commutative algebra is Grothendick's work on the elements of algebraic geometry. This is maybe 2,000 pages long and contains several hundred pages of background commutative algebra spread out through the various volumes. Finally, I will just finish by mentioning a historical book. This is the Algebraic Theory of Modular Systems by Macaulay. This may be, as far as I know, it's the first book that was actually written on commutative algebra. It's a little bit tricky reading it because all the terminology has changed. For instance, modular system is the name for what is now called an ideal of a ring. So I just say give some administrative information about this course. First of all, the full list of lectures for the course should appear in a playlist labeled commutative algebra on the channel. I'll try and put a link to it. The link to it may be a small white circle, possibly somewhere up here, if I managed to figure out how to do this, only it may not be because YouTube seems to change the way it does links every year or two. So by the time you see this, this may be out of date. The white circle or whatever YouTube has at that time should also have a link to the next lecture in the series. I should perhaps also mention, if you want an online reference for commutative algebra, there's something called the Stax Project, which if you want to see, you will have to search on Google because I haven't figured out how to put working links on YouTube and in any case, the link will probably change. So the next lecture will be a quick review of basic ring theory like rings, ideals and modules.