 In this video I want to talk about the characteristic of a ring with unity and so to do so consider for the moment any ring whatsoever under the sun. Given any ring you can take the ring of integers and it acts upon that ring. That is you can construct the idea of an action so an integer acting on a ring thus producing back a ring element by the following rule. If you have a positive integer like 1, 2, 3, 4, etc, then we take the action in dot r to be the sum of our end times. So you have r plus r plus r plus r plus r plus r all the way up to doing it in times. I should also mention that if you take zero dot r, this is understood just to be zero. And if you take a negative number, negative n dot r, this is defined just to be n dot its negative. So you just add together the negative inverse of that number so you get a negative r minus r minus r and you do that in times. So we can make sense of what it means to have an integer acting on a ring. Well, what do you mean by action? We had an accent for a group action. What does it mean here? Is this like some type of ring action? And the idea is actually exactly that. We do have a notion of a ring acting upon an abelian group, which every ring itself is an abelian group, so a ring can act upon a ring. Much of the same way that a group can act on a set, a ring can act upon an abelian group. And sometimes when we talk about group actions, the set we're acting upon is itself a group. Well, the same thing happens with ring actions for which when you have a ring action, a ring necessarily will have to act on an abelian group, but that abelian group itself could be a ring, so there could be more algebraic structure there. And so I'm not going to list all of the axioms of a ring action, but basically I'll push you back to your days of linear algebra. That scalar multiplication on a vector space is an example of a ring action. For a vector space, you do require that the ring that's acting is a field. And the way that a field acts on a vector, that's basically the axioms of a ring action. So I don't want to go through all of them, but just some of them I'll remind you like one acts on it to give you back R. We also have, if you have two integers m and n, which are acting on an element R. This is going to be the same thing, basically a distributed property m dot R plus n dot R. We have it so that n is acting on the sum of two things, two ring elements, r and s. This becomes n dot R plus n dot s. And there's several more axioms there. There's about eight of them, I believe, off the top of my head, that a ring action has to satisfy to actually be a ring axiom. The one that actually care about for this conversation is the following principle right here. The other ones are somewhat irrelevant for us. That is, if we take, so this is actually the case if you have a ring, that the set we're acting upon is a ring, not just an abelian group. If an integer acts upon a product, this is actually the same thing as n acted upon the first factor. And this is just a consequence of the distributive law for the ring. So if you take n dot R times s, well, n dot R means you add together n copies of R, for which s then distributes onto each of those. So we're going to get n copies of R s added together. So that's exactly what n dot R s is all about. And so because of this, because of the action axioms and particular this behavior, kind of like an associative law with regard to the ring multiplication and the integer action, we often drop the dot entirely and we actually think of it as a product. We think of it as a coefficient. So we actually talk about like, oh, 2x plus 3y, things like that. 2x just means x plus x. That's available in every single ring that we could consider. So now we get to the titular topic for this video here. Suppose R is a ring. We then define the characteristic of that ring to be a positive integer. Commonly it's denoted as char of R there, the characteristic of R. It's equal to n, where n is going to then be the smallest positive integer such that n times R is equal to zero for every element of the ring. So that is to say the integer n acts on the ring by being zero. So n times R is equal to zero for every element of the ring. That's if there is such a positive integer. It could be that there is none. If there's no positive integer, we actually call the characteristic of that ring zero, rings of characteristic zero. We've seen several of these before. So for example, the ring of integers, the ring of rational numbers, real numbers, complex numbers, these are all rings of characteristic zero. There is no finite number for which we add elements together, you get back itself. But if we consider, say, the ring Zn, the ring of integers mod n, it's kind of in the name there. This is a ring of characteristic zero because if I take something like 2 plus 2 plus 2 all the way down, if I do it n times, well, as an integer, this is the same thing as 2n, which is then congruent to zero mod n. Therefore, Zn is a ring of characteristic zero. And I also want to throw in a plug for matrix rings here because using a matrix ring, we can actually create a new ring whose characteristic will match up. So recall the notation M sub N by N of R. This is the set of N by N matrices whose coefficients, whose scalars come from the ring R. Well, the matrix ring, it's a ring, of course, it'll have the same characteristic as the coefficient ring has. So if it's a ring of characteristic zero, let's say the real numbers, the complex numbers, the matrix ring will be characteristic zero. But if you take the matrix ring over like mod 2, Z2, that matrix ring will also be a ring of characteristic two. And so the characteristic of a matrix ring is just reflective of the characteristic of the coefficient ring. So in some regard, we can make lots of different rings of the same characteristic non-isomorphic to each other. All right. Now when it comes to rings with unity, the calculation of the characteristic is a lot easier. And this is going to ultimately lead to an observation we make about integral domains. So if R is a ring with unity, that is, it has a multiplicative identity, then the characteristic of the ring is going to be the additive order of that unity in the Abelian group R, plus. For which I should make mention that the characteristic is going to be the smallest positive that makes, it's essentially the characteristic of a ring is the additive exponent in the Abelian group R, plus. That is to say it's the smallest positive power that makes everything go to the additive identity, ak zero. So in particular, if I take n times R, that's always equal to zero. So n times one, of course, is equal to zero. What we're trying to say here is you don't even have to look at anything else. Whatever happens to the unity is what happens to everybody. Okay. So let's let's make a comment about that. Well, I should say let's look at the proof of this thing. If one has an additive order of n, then n is the least positive integer such that n times one is equal to zero. Okay. Take an arbitrary element of the ring R. Let's look at n times R. Well, n times R is the same thing as n times one R. Okay. And then by the associative property for this ring action, n times one R is the same thing as n one times R. But like we have by assumption n one is equal to zero. So we get zero and anything times zero in a ring gives you back zero. So if n sends one to zero, then n will send everything to zero. And so in particular, as there's no smaller power, I should say no smaller exponent. No, that's the wrong word too. No smaller coefficient, excuse me, of one that gives you zero. Then that shows that the characteristic of the ring has to be n. If such a thing exists, of course, if it's characteristic zero, that means no power will send one to zero. And so that's a characteristic zero ring. All right. So that's pretty cool. If you have a ring of zero, ring of unity, excuse me, ring with unity, you can check the characteristic by just looking at the unity. How many times do you have to add together the unity? How many times you have to add together one to give back zero? If that happens after finitely many steps, that number of steps is the characteristic. If adding one together never gives you zero, then you have a ring of characteristic zero. At least that's how easy it is to do when you have unity. All right. So let's get to the idea of a domain. So the characteristic of a domain is always zero or its prime. And remember, a domain here is we don't assume it's commutative, but we have a ring with cancellation and we do have unity always assumed for in a domain. All right. So let's suppose that it's not zero. So okay, zero is a possibility. Let's not consider it. So if the characteristic is not zero, there's some positive number so that everything, if you add that element together in many times, you get back zero. In particular, since domains have unity, we have that this number n, when you act on one, when you add one together in times, you get back zero. Okay. Now we claim that this number is a prime. So for the sake of contradiction, suppose it's not a prime. Okay. Suppose n is a composite integer. There are two positive proper divisors, a and b, such that n is equal to a times b. Okay. So now consider the equation we had before. We have zero is equal to n times one. n is of course the same thing as a b, for which we can write this as a times b one, like so. We can also then factor that even better. This is a b times one is the same thing as a one times b one. So that, that has to say, we're just adding together, you know, if we break this thing up, we're just adding together one plus one plus one. And we do this, you know, a b times both of these things do the exact same thing. All right. But because a and b are both between one and n, we're going to get that a one is not zero and b and it's not zero. Okay. Because this is something smaller than the characteristics and is the smallest coefficient that makes one go to zero. So a one can't be zero b one can't be zero. But then look, we have a product of two non zero values that equals zero domains can't do that we get a contradiction. So in a domain, the characteristic of the ring is always zero or it's always prime. And this is a very useful observation as we continue to study domains and factorization into the future. So that brings us to the end of lecture 12 here. Thanks for watching. If you learned anything about integral domains or characteristics, please like this video, subscribe to see more videos like this in the future. And as always, post your questions in the comments below. And I'll be glad to answer them as soon as I can. Bye everyone.