 Welcome back to our lecture series, Math 4220, Abstract Algebra I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In this lecture 37, we want to talk about the idea of rings, right? What is a ring after all? I'm not talking about like Frodo's ring of doom or anything like that. Ring has a very proper algebraic definition, just like groups and other things we've talked on this lecture series. In fact, up until now in our lecture series, we focused on algebraic sets with a single binary operation with very few exceptions. Now, often in algebra, it is useful to consider a set with multiple operations that interact with each other in a very nice way. For example, vector spaces. In the last couple of lectures in this lecture series, we've talked about matrix groups, which really drew us into the realm of linear algebra. For a vector space, there are two operations in play. You have this idea of a vector addition, which is an operation maybe from like fn cross fn to fn. You add together two vectors and you make a vector, but then there's this other idea of scalar multiplication for which you multiply a scalar by a vector and you get a vector. These two operations need to really work together in some compatible way in order to even make it worth discussing the two objects together. With respect to addition, a vector space makes an abelian group. But why do we care about scalar multiplication if it's somewhat irrelevant to how addition operates? Well, the thing is when it comes to scalar multiplication, scalar multiplication distributes over vector addition. If you have r times v plus u, this is equal to rv plus ru, things like that. The scalar multiplication distributes over vector addition. We need that there's some type of compatibility condition between the two operations. It's not just studying two operations in isolation. We're studying the two operations together because they are compatible with one another. But that's not the only place to stop with vector spaces. If you take an inner product space, for example, that gives us a third operation called the inner product, the most famous inner product versus the dot product. Now the inner product also needs to behave nicely with the other two operations. So if you take the dot product, like take u dot, let's say v plus w, the dot product also distributes over vector addition. So you get something like u dot v plus w is u dot v plus u dot w. We also need that the dot product behaves well with scalar multiplication. If you take ru dot v, this is the same thing as u dot rv, which is the same thing as just r times u dot v. So the operations in play, the more and more operations you add to your algebra, they need to somehow mingle well with one another. There needs to be some type of compatibility going on, which could be some type of like distributive law or homogeneity property or something like that. Now another example we've talked about in this lecture series is we've seen that with the algebra of sets, we have things like intersections and unions and set differences and such. This is an algebra with three operations, in which case, you could also think of like complementation instead of set differences, you could take compliments, what have you? How do these things interact with each other? Well, you have distributive laws, that is intersections distribute over unions, unions distribute over intersections, in which case you also have the demorgan laws that connect set compliments with unions and intersections and such. There's all these conditions that allow the operations to co-mingle with each other. So the study of algebraic rings is in this same vein. That is, a ring is loosely speaking an algebraic set with two binary operations. And those two operations are always referred to as addition and multiplication, for which we'll use the standard plus symbol to denote addition, like you see right here. And for multiplication, we'll typically use juxtaposition for multiplication, but if we need to emphasize it, we might use like a little dot or a cross or something, the usual multiplication symbols here. And so we then for a ring, get the following eight axioms that are required for these binary operations. The set we'll call R, we have these binary operations. So again, addition is a binary operation, it's gonna be R cross R to R, right? Multiplication is a binary operation, we're gonna take R cross R to R, binary operations in their usual sense. So what do we require for these operations then to be a ring? We first require that addition be associative. That is to say that for any three elements of the ring, R, S and T, we have that R plus S plus T is equal to RS plus T. So we can re-associate with addition. The second axioms require that addition also be commutative. That is, for any elements R and S inside the ring, we have that R plus S equals S plus R. Whenever we use additive notation to describe an operation, it's always inferred that the operation is addition. Excuse me, the operation is commutative. If you ever had some type of non-commutative addition, you might be exiled from the mathematical community because it's just too taboo. Addition should always be commutative. Thirdly, we require that addition have an identity element. There must exist an element which is typically called zero, nearly it's always called zero inside of a ring, the additive identity. There exists an element zero inside the ring. So that's that if you take any other element in the ring, we have that R plus zero is equal to zero plus R, which is equal to R. So we have an additive identity. And then fourthly, we require that there be additive inverses. So for all elements R inside the ring, there exists an element which we call negative R. So we use negative notation to represent the additive inverse. So I don't mean negative R is negative one times R because up to this moment in the axioms, what does negative one even mean? We don't know. Negative R is just notation to describe the additive inverse of R. So for any element R, there is a negative R, so set R plus negative R is equal to negative R plus R, which equals zero. So if you look at just the first four axioms of a ring, this right here tells us that the structure, if we just look at addition, just addition, R plus this is gonna be an Abelian group. So addition in a ring always forms an Abelian group. It's commutative associative identities and inverses. Okay, so that's the type of structure that addition will have inside of a ring. What type of structure does multiplication have in a ring? Well, you get the fifth axiom, which is gonna be multiplicative associativity, which as we saw a moment ago with an additive associativity, since multiplication is associative and we take any three elements inside the ring, RS and T, we require associativity with multiplication. So we get R times ST is equal to RS times T, which I'm using the dot to emphasize multiplication right here, but typically we write this as R times ST is equal to RS times T, just the juxtaposition there. All right, and so when it comes to a ring, that's the only requirement we have on the multiplication, that multiplication is associative. In which case then, if you look at just R with the binary operation of multiplication, this gives us what we call a semi-group. We have this binary operations as associative, we do not have any requirement of inverses, we have no requirement about identity. Now, we often do care about rings that have extra structure on multiplication, but in those general sense, a ring only we require associativity. Now I do have to caution the viewer right here that there are some authors of algebra textbooks that do not even require associativity for multiplication in a ring. Now that sort of looser definition of a non-associative ring is not as common, especially not at the undergraduate level. I think it's quite universal that for undergraduate abstract algebra courses, rings are assumed to be associative, the multiplication. And if you don't specify, most people will infer that the ring multiplication is associative. If you want non-associative rings, you do need to specify that because by default that's not the case. So okay, the operation of addition forms an abelian group. The operation of multiplication forms a semi-group, right? So not a lot of structure is required for the multiplication, but like I was saying at the beginning of this video, we can't just stop there, we can't just have an abelian group and a semi-group. They have to somehow interact with each other. And in rings, that interaction comes from the distributive laws. And these are laws that we're probably quite used to, right? You have left distribution and right distribution, for which case the left distributive law says that for any elements R, S and T, if you take R times S plus T, that's the same thing as RS plus RT. We also have the right distributive laws would say that R plus S times T is equal to RT plus ST. So we can distribute on the left and the right in the usual sense. Now I do need to specify that when it comes to a ring, we do not assume that the multiplication is commutative. And therefore we do actually need a separate left and right distributive laws because if you had just left distribution and no community axiom whatsoever, you can't prove right distribution. You can have rings that are left distributive but not right distributive and vice versa. So we do need both distributive laws in order to make this thing work. Okay? Now, like I said earlier, that there's not a lot of axioms placed on the multiplication of a ring. We just require it's associative but maybe we want some extra axioms as well. And so furthermore, we could say that a ring is called a ring with unity if R is a ring which satisfies the additional axiom of a multiplicative identity. So if you have a ring which has a multiplicative identity, we call this a ring with unity. Unity here is a term used to describe the multiplicative identity of a ring. And following the convention we had before, that multiplicative identity is nearly always called one. The additive identity is zero, the multiplicative identity is equal to one. Now I do have to specify that when we talk about a ring with unity, we are assuming that the multiplicative identity that is the unity is distinct from the additive unity that is the additive identity, which is zero, right? We don't want one and zero to be the same thing. The only potential counter example, if we allowed zero to equal one, you would get the so-called zero ring, which the ring contains only one element which we call zero for which addition is obvious you get zero plus zero is equal to zero. We would also get that multiplication is obvious zero times zero has to equal zero for which that satisfies the normal condition that satisfies all the axioms above in a very trivial manner, right? This will be a ring, this is called the zero ring, right? For which the zero ring satisfies the eight axioms of a ring, it also satisfies axiom H here, it has a multiplicative identity will be zero. We don't call this a ring with unity though because we want to distinguish the zero ring from other rings with unity. We want the one and zero to be distinct elements. So the zero ring is excluded. And if we have a multiplicative identity, the unity here, it needs to satisfy that for any element R inside the ring, R times one equals one times R, which equals R. Additionally, we couldn't want that the multiplication of the ring be commutative. So we talk about a commutative ring. A commutative ring is a ring which satisfies the extra axiom of multiplicative commutivity that is for all elements of the ring, R times S is equal to S times R, all right? And so when you use the adjective commutative to describe a ring, you're describing the multiplication because with addition, addition is an appealing group, it's commutative. So if we talk about a commutative ring, we're saying multiplication is commutative. If we talk about rings with unity, we're saying that the multiplication has an identity because of course the addition will always have an identity. You can also combine these two together and you can talk about a commutative ring with unity, with unity here. So this would be a ring which the additive structure of course an appealing group, the multiplicative structure would be a commutative monoid because it has associativity identities and commutivity there. So it might be tempted that then to ask, well, could we also add the axiom of inverses to the multiplication? It gets a little bit more complicated because of how the distributive law looks. And we're gonna talk about how you can have inverses inside of multiplication. That is when is division possible. We'll talk about that in the next lecture or lecture 38 in our series. So we're gonna stick with just rings, commutative rings, rings with unity and commutative rings with unity for the time being. So let's consider some examples of rings that we've run across previously and either this lecture series on abstract algebra or perhaps previous algebra settings. So we've talked about the group structure of Z, Q, R and C with respect to addition. But these sets also have multiplication, multiplication that does distribute over the addition that we have. And so these are all examples, Z, Q, R and C are all examples of commutative rings with unity. Z has unity. It's the number one and it distributes, right? So you get Z, Q, R and C, these are all commutative rings with unity, right? They have a one, they have the multiplication, commutes, multiplications, associative, we have distributive laws, all that stuff. On the other hand, if you take the set of natural numbers, we know that the set of natural numbers cannot be a ring because as we've talked about with the group structure of addition, the natural numbers with respect to addition are not an abelian group because they're lacking inverses. Addition is commutative for the natural numbers that is associative. You have zero, the natural numbers include zero here. So you almost get an abelian group. With respect to addition, the natural numbers forms what we call a commutative semi-group, specifically it's a monoid because it's associative, there's an identity, it's commutative, but we don't necessarily have inverses whatsoever. Now, when it comes to N being a ring, the only thing missing is the additive inverses because addition is associative, commutative, you have zero, multiplication is associative, commutative, it has even unity, right, has the number one. The distributive laws hold the only thing that's missing here is that it doesn't have additive inverses. And so if you take the previous eight axioms of a ring and you remove the condition that addition has inverses, you actually get something called a semi-ring for which the natural numbers form a semi-ring. And in fact, this will form a commutative semi-ring with unity. So it's really close to being a commutative ring with unity, but you don't have inverses for most of the natural numbers, in fact, because you don't have negative numbers. That's the main difference there with the natural numbers as close to the integers. So the natural numbers are not a ring, it's a semi-ring at best. So let's next consider some subgroups of the integers, right? So if you pick your favorite integer, then the usual notation, Nz, will represent all the multiples of N inside of z right there. So for example, 2z, this would be all the even numbers, so you're getting like negative six, negative four, negative two, zero, two, four, six, eight, right? Who do we appreciate? So 2z is just the set of all even numbers, 3z would be all the multiples of three, 4z is all the multiple of four. We've seen this notation before. Now it turns out that Nz is in fact a commutative ring, but it doesn't have unity, assuming N is not plus or minus one. If it's plus or minus one, this is just the ring of integers as usual. So let's think about it. In terms of addition, this is going to be an amelian group. In fact, it's a cyclic group. With respect to multiplication, multiplication is associative because it's just integer multiplication. Multiplication is commutative because it's just integer multiplication. It's distributive because again, this is just the operations of usual integers inside the field of real numbers. What's lacking though is an identity, a multiplicative identity that that is the unity, right? Because if you take these numbers here, like the product of any two even numbers is going to be even, that part's still the same. Distributive, associative, commutative, we get all of that stuff for the operations of addition and multiplication. But there is no number, right? There is no even number, which if I take two times x, I'm gonna equal two, right? There's no even integer that does that because in the integer system, right? If you're thinking about this in the integer, you could cancel out the two and you get x equals one, but one is not inside of two z. So there is no even integer that acts like a unity here. So this is a commutative ring without unity. Let's look at another example of rings that we've run across previously in our lecture series here. If you take your favorite integer n, then consider the set in z, zn, excuse me. And so I guess we probably should specify that n is actually a positive integer in this situation. The set of congruence classes mod n forms a ring under modular addition and modular multiplication, which we've talked about these things in isolation, right? We've talked about the group nz, n, excuse me, comma plus, right? So this would be a cyclic group of order n. We've also talked about the group zn star with respect to multiplication where the star, we just take those elements which have multiplicative inverses, which have reciprocals, because that makes a group. But we could relax and talk about nz, multiplication, right? This wouldn't make a group anymore with respect to multiplication because not every element has an inverse, but that's okay in a ring. We don't require that every element has a multiplicative inverse. So if you put modular addition and modular multiplication together, you're gonna form a commutative ring with unity. And this then gives us an infinite family of finite rings.