 Welcome back to our lecture series Math 4,230, abstract Algebra 2 for students at Southern Thai University. As usual, I'll be your professor today, Dr. Andrew Missildine. In this lecture, number 12 in our series, I want to talk about the idea of a unit for which if you're not sure what that is in ring theory, a unit is going to be an element of a ring which has a multiplicative inverse. It's important to recognize that unlike groups, multiplication and rings, it doesn't necessarily have inverses. I mean, hey, one of our assumptions about a ring is that it may or may not have even a multiplicative identity. We talk about rings versus rings with unity. If there's no unity, there's not going to be any multiplicative inverses, a.k.a. there's no units, right? So even if we have unity, there's no guarantee for a generic ring with unity that we have multiplicative inverses. And as such, the consequences of a version in group theory don't always carry forth into ring theory. And so there's some things I want to mention that are on the screen right now. In particular, in group theory, one of the results we get because of inverses of elements is this idea of cancellation. So cancellation usually looks something like the following. If A, B, C are elements of a set such that if B, A equals C, A, that implies that B equals C. So that's what we typically call right cancellation. And then similarly, if A, B equals A, C, that implies B equals C. OK, so this was a theorem we proved in group theory back in algebra one. But in general groups, excuse me, general rings do not have this property. So in this context, I'm actually referring it to it as a cancellation axiom. And it could be that you have an algebraic structure with binary operations, we'll call it, we'll call that binary operation multiplication for the lack of a better term here. Does your operation satisfy the cancellation axiom or not? Groups always do because of the inverse axiom. It implies cancellation. But in the general algebraic setting, you might not have that. Honestly, semi-groups, which is a generalization of the idea of groups, they don't have cancellation in general. But quasi-groups, a different generalization of groups, this is sort of like their thing. A cancellation axiom is like what makes a quasi-group, a quasi-group. Then, of course, we're focusing in rings in this conversation. In rings, cancellation is not guaranteed. And so in this lecture, as we talk about the idea of an integral domain, we'll see that integral domains are really those rings with the cancellation axiom as satisfied. And I want to point out one of the main reasons we can't do cancellation in a ring is actually because of this idea of dominance. Sometimes it's called absorption, where if you have an algebra, so that is there's some set with a binary operation attached to it, maybe there's multiple operations, but we'll just consider one binary operation for a moment. We say an element is dominant. If you take that element, excuse me, we'll call the dominant element zero here, thinking of ring theory. So if you take any element of the set, any element of the algebra, zero times that element is always equal to zero. And that element times zero is always equal to zero, because we're not assuming it's commutative necessarily. So this in some respect is like the opposite of the identity. If we're thinking of like multiplication here, what's a multiplicative identity? A multiplicative identity would mean that one R is equal to R1. So it commutes with everything. But then that's equal to the other element. So a dominant element is kind of like Narcissus, you know, he's always thinking about himself. He's always reflecting back to him. Every product was zero, give you back zero. The identity is more like echo. Echo is always thinking about someone else, or in particular, she's always thinking about Narcissus, right? To continue with the Greek myth there. So a dominant element is like the opposite of an identity. In particular, if you have a dominant element, then you cannot have universal cancellation. Because of every element times by zero gives you back zero. If there's two non-zero elements, even if there's one non-zero element, right? If R doesn't equal zero, that means zero R equals zero times zero, which is equal to zero. You can't cancel the zeros and get back R, right? You couldn't tell the difference. So in rings, the zero element, which is why I denoted it like this in this axiom, in a ring, a ring always satisfies the dominance principle. Zero is always an absorbing element, all right? And so this is a theorem we prove for ring theory. I've listed it as an axiom because there do exist algebras, which would assume dominance as one of its fundamental axioms. We're not going to delve into that too much right now, but just like how cancellation is a theorem in group theory, dominance is a theorem of ring theory. And so rings always have a dominant element and it's the additive identity, which we typically could call zero. At that, I should say it's dominant with respect to multiplication because it's the additive identity. It's a consequence of the distributive law. But this idea of cancellation goes beyond just zero, right? I want you to consider the ring, mod 6, of integers, mod 6, Z6 here. Consider the following equation. Three times four, that's equal to 12, which is congruent to zero, mod 6, but also take three times two, that's equal to six, which is also congruent to zero, mod 6. So when you look at that equation right here, three times four and three times two, you can't cancel the threes because four doesn't equal two inside of this ring. So this issue of cancellation is beyond just the dominance of zero. We have things like this going on right here. You can't cancel three in the rings, Z6. But it turns out, and we'll talk about this in just a second, that the reason we can't cancel three is actually because it's related to six. In particular, three times two is equal to six, and six is zero, mod six, of course. We'll delve into that in just a second, right? But let's look at some matrix rings for a second. So imagine that R is just any ring and little R is any non-zero element in the ring. OK, if I take two by two matrices, so let's take an R in the one-one position, zeroes everywhere else, we times that by the matrix, which has an R in the two-two position, zero everywhere else. When you think of the multiplication, you take this row times this column, you're going to get zero plus zero, which is zero. You'll take the first row times the second column, that's still zero plus zero, which is zero. Then you take this row times that, you get zero plus zero, which is zero. And then the second row times the second column, you get zero plus zero, which is zero. So the product of these two rings, excuse me, the product of these two elements is going to give you back zero. But in fact, I could replace this with a whole lot of other things, right? Why doesn't have to be R and R? I could put R and S and we get the exact same thing again, right? In particular, if you set that to be zero, there you go. It's like you can't cancel this matrix because I can't tell what's happening here. You might be able to say something about the matrix, but it's like, whoa, there are lots of different matrices. In fact, I have a whole, I could replace little R with any element of the ring. And so no cancellation can happen. At least I should say that cancellation doesn't happen in general, inside of a ring. Now, there is one important place where cancellation can happen. Say that R inside of a ring is a unit. Remember, unit means it has a multiplicative inverse. That is to say R times R inverse is equal to the identity, multiplicative identity, we'll call it one in this situation. Because if you're in that situation, let's say R X equals R Y. OK, who cares about R, who cares about X and Y here? But just think about R, if I times both sides of the equation by R inverse, R inverse here by associativity, we redo it, R inverse times R times X. And then you're going to get R inverse times R times Y. R inverse is one in both situations. We have one X equals one Y and therefore X equals Y. So if you have a unit, we can cancel units using their multiplicative inverses. This is the exact same proof we used in group theory in abstract algebra one. So units are elements that can be canceled. But a curious thing in rings is that we can have elements which are not units. That is, they don't have multiplicative inverses, but still can be canceled. This cancellation is sort of a weaker condition than having a multiplicative inverse is inverses imply cancellation. But cancellation can happen without being a unit. And that's the topic we want to explore a little bit more right now. So now we approach the main topic for this video. Suppose that R is a ring and take two elements inside that ring called little R and S. We say that R and S are proper divisors of zero if, first of all, they are nonzero values, OK, so R and S themselves are not zero, but their product is zero. So R times S is zero, but neither of the elements R and S are themselves proper divisors of zero. Now, why do we say proper divisors of zero? That is the full term proper divisors of zero. Sometimes people call these divisors of zero, you drop the proper. Sometimes we could these are called zero divisors. What do we mean by here in this situation? Well, I should mention before I finish this definition, the adjective proper is used here because, you know, if you did have R equals zero or or if S equals zero, then, of course, R S equals zero. That's always the case, right? If you have zero times S, that's always equal to zero. So in some respect, every number is a divisor of zero in a ring because you can times it by zero to give you back to zero, right? So proper is significant here that, you know, it's not zero is not involved in the product or I should say zero is not the factors involved, but you still get a product that's equal to zero. And so, like I said, again, some people call these, did this drop the word proper? Every element of a ring is an improper divisor of zero. That doesn't mean much. It's the proper divisors of zero we care about. But because we only care about the proper divisors of zero, sometimes people just drop the term proper and talk about divisors of zeros or zero divisors, and that's what that's always to be understood as. OK. All right. So zero divisors are elements of a ring, which are nonzero, but whose product is zero. Then we say that a ring with the unity is a domain if it has no proper divisors of zero. So there is no product of elements that give you zero without using zero as an element itself. When it comes to a domain, we always do require it has a unity. And then, of course, it has no zero divisors. A commutative domain is typically called an integral domain. And so one has to be careful in the literature that some people, when they talk about domains, they really mean integral domains because they're so entrenched inside of commutative algebra. That is the study of commutative rings that the thought of having a non commutative domain never comes to their mind. So they just get sick of saying integral domain. So they say domain. But for those who study non commutative rings, well, you often are interested in domains that are not commutative. So you talk about domains versus integral domains, that is the that's the dictionary we will use in this lecture series. A domain could be commutative, it might not be commutative, but an integral domain will always be commutative. But if we talk about a domain, we're making no assumptions about whether multiplication is commutative or not. So speaking of axioms that we've introduced into this lecture about dominance, we talk about cancellation, the non existence of proper divisors of zeros can actually be axiomatized. And that's what would be called the zero product axiom. So I'll use language of a ring, because this is where we really care about it, because after all, you need to have a zero element to even talk about zero products. So when we talk about zero product, we're really talking about a dominant element. Our multiplication has some dominant element, which again, as we're studying rings, we'll always call that element zero. And so is it possible to have a product of two elements equal the dominant element without one of the factors already being that dominant element using zero? Specifically, the zero product axiom would be that two elements of the ring, if their product is equal to zero, that means one of the two factors was equal to zero. A, B can only equal zero if A equals zero or B equals zero. In other algebra classes, this is sometimes referred to as the zero product property. This is a property of the real number. So like many college students or high school students as they're doing early algebra classes, they learn about the zero product property. They use this to solve polynomial equations. Right. So if X minus two times X plus one equals zero, excuse me, that means X equals two and X or X equals negative one, because you can set with only with the product equal to zero is one of the factors equals zero. And therefore you consider the two possibilities will maybe X minus two equals zero, that's where the two comes from, or maybe X plus one equals zero. That's where the negative one comes from. So the zero product property and like college algebra is a useful technique to help you solve a variety of equations like polynomial equations using factoring. So this is a this is a property that's nice to have in a ring. So given that the definition of a the definition of a ring doesn't actually prevent such a thing, in fact, we saw on the previous slide examples of the product to non-zero elements that equals zero, like think of those two by two matrices, there are rings that don't satisfy the zero product property. So sometimes we have to add that axiom to the axioms of a ring in order to guarantee it. And with that perspective, that's exactly what a domain is. A domain is a ring with unity that also has the zero product axiom. OK, if you have no proper divisors of zeros, then you satisfy the zero product property if and only if. So domains are exactly rings with unity with the zero product axiom, in which case then an integral domain is just a domain, which then you throw on the commutivity axiom. So we can stack these axioms on top of each other to make more structured rings. I should mention that the rings, Q, R and C, these are all examples of integral domains that are commutative because they satisfy the zero product property. For the same reasons I was talking about here in College Algebra, these are the rings you're looking at, Q, R and C. They satisfy the zero product property. In fact, these rings, Q, R and C, they're actually fields, right? And so every element other than zero has a multiplicative inverse. Every non-zero element is a unit. So if ever you have a product of two elements, R, S that equals zero, if R and S are not zero, then you can multiply both sides on the left by R inverse, right? R inverse times zero is going to be zero. But R inverse times R is the identity. It's just going to be one. And so that means R inverse times R, S is going to equal S. So if R is not zero in a field, the other one had to be zero. And a similar argument also works for any skew field. So every field is an integral domain because every non-zero element is a unit and a skew field, which is not necessarily commutative, but every element still has a multiplicative inverse. That's also going to be a domain by the same reasoning. So we've seen so far that units imply cancellation. Units also imply the zero product property. So if you fail the zero product axiom or you fail the cancellation axiom, it turns out that this is because of some lack of inversion, some lack of units here. Another example of an integral domain is the ring of integers. Now, this is a very important example because the integers do not form a field. In fact, most integers do not have multiplicative inverses. The only ones that do are the unity, which is its own inverse. And it's additive inverse, which is also its own multiplicative inverse. And those are the only two integers that are units. But Z here is still also an integral domain. The only product of integers that gives you zero is because one of the factors are zero. And the easiest way to see that is that the integers live inside of a field. They live inside of the rational number. So any product of integers that equals zero is also an equation, it's also a factorization of rational numbers. So you could use the rational inverse of N to show that M has to equal zero or something like that. I mean, that's a topic we'll talk about later on about extending an integral domain into a field, but that's that's a topic for another day. All right. But I also I really want to mention this example because this example really is the reason why they're called integral domains. Because after all, integral, not the calculus sense. Integral is the adjective form of integer. So if something is integer like, we can call it into integral, right? So an integral domain is a ring that essentially behaves to some degree like the ring of integers. The ring of integers is the archetype for integral domains. And that's where the name comes from.