 Welcome back to our lecture series, Math 42-20, Abstract Algebra 1 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In this lecture 38, which is the final lecture for our series here, we're going to talk some more about rings, which we had introduced previously in our series. In fact, in this video, I want to talk about the idea of a group ring, which essentially generalizes the idea of a polynomial ring which we had introduced in the last video, in fact. With polynomials, we had learned that because of the distributive property, there's only one way that you can multiply together polynomials. It's basically by the usual FOIL method. If I had something like, let's say, x cubed plus x squared, and then I multiply that by say, x minus x to the fourth or something like this. If we have to FOIL this out, by the distributive law, you can distribute the first ones through so you get x cubed times x minus x to the fourth plus x squared times x minus x to the fourth. That's how the distributive law works from the left, and then I should say from the right distribution, and then we're going to go the other way around like so. I guess I should say it like this. We're going to distribute this way, because what we had done previously, we distributed this onto the two pieces. That's what we were doing. Now, if we use left distribution, we end up with x cubed times x minus x cubed times x to the fourth. We get x squared times x minus x squared times x to the fourth. And so what we have in front of us is we're multiplying out these polynomials. You'll notice I chose all of the coefficients to be plus or minus one here, because I didn't want us to worry about the coefficients. How you multiply the coefficients is determined by the coefficient ring R. But how you deal with the monomials is really what I'm concerned with right now. Now, by the usual rule, when you multiply this powers of x, you're going to end up with x, you just add the x most together, x to the fourth, x to the seventh, x cubed, and then x to the sixth, and then combine like terms if possible. That is, at the end of it, we have to decide how do you multiply together the monomials, right? For which each of these monomials, you add together the powers, three plus one gave us a four. We had three plus four gave us a seven. We had two plus one gave us a three, and then we had here two plus four. So we added together the exponents. What if we combined the monomials together by a different rule? What if in fact the rule comes from, not necessarily the group of integers under addition, what if the rule comes from any type of group, right? And this then leads to the idea of a so-called group ring. So we can generalize the construction from polynomial rings to any arbitrary group, right? So let R be some ring. This will act like the ring of coefficients, and G is gonna act like, well, G's gonna, it's a multiplicative group. It's gonna act like the monomials that we build polynomials from. And then with this, with a ring and a group, we define the so-called group ring, super clever name, I confess there. The group ring will be denoted R of G, in which case we take the sum of all linear combinations of elements of the group. And as the group is considered a multiplicative group, we don't add together elements of the group except for we add together like terms. So let me give you an example of such a thing. So let's consider the group ring for which we're gonna take integer coefficients, and as elements of the group, we're gonna take the dihidra group D4, okay? So some examples of elements in this group would be things like one plus R plus R squared. We could take two minus three R plus five S. We could take, say, four R squared minus two S, plus let's say seven R cubed plus four times R squared S, R squared S, things like that. So we think of, we're thinking of these elements here, these polynomials. They're gonna look like polynomials to any other sense, right? You know, if you were to present this polynomial to a college algebra student, it's like, okay, I think we have some polynomials in terms of the variable R and S, right? That's how they interpret it. And so for the Morse part, that's exactly how you see it. When it comes to addition of elements of a group ring, we add together like terms, just like we would polynomials. That is, if you have an arbitrary element of the group ring, it's gonna be a linear combination of elements from the group. So the group acts as your monomial, the group elements. And then the coefficients come from the ring. If you add together one group polynomial with another group polynomial, you just add together like terms. So for example, if we added together these two group polynomials, we get one plus two, because that's a like term. So we're gonna get three. We're gonna get R minus three R, so we get a negative two R. We have an R squared that doesn't find with anything and we have a plus five S that doesn't combine with anything else. And so that's how we add together these elements. We just add together like terms, right? So we just treat these monomials as their symbols, as their variables, as they're these indeterminate elements. We just add to R's with R's, R squared with R squared, S's with S's, that's all that we have to do. Now, so in terms of addition of these elements in a group ring, you just add together like terms. But what if we wanted to multiply together elements of this group ring? Well, basically, using the distributive law, there's only one way that you can do it, like we saw with the polynomials earlier, but it's gonna be determined by now the group relationship, the group product. So if I took, for example, one plus R plus R squared and you multiply that by two minus three R plus five S, basically what you're gonna have to do is you're gonna have to multiply this out by the distributive property. So you're gonna get one times all of these elements, which is gonna give you one times two. You're gonna get here one times negative three R, you'll get one times five S, that's the first distribution that you have to distribute R through all of these things, for which then you're gonna get, when you do that, you're gonna get an R times two, you're gonna get an R times negative three R, then you're also gonna get R times five S, okay? And then likewise, you're gonna distribute that R squared through, in which case you end up with an R squared times two plus an R squared times negative three R, and then lastly you're gonna get an R squared times five S. In which case we do all of these products now, for which you're gonna multiply the coefficients together, so you get one times two, which is two, for the next one you get one times negative three, this is negative three R, one times five, you're gonna get five S, like so. But then whenever you have to multiply like a group element times a scalar, well, that's just gonna be the coefficient, right? So you're gonna get a two R, you're gonna get on the next one R times negative three R, you take one times negative three, which is negative three, but then you get R times R, which in the dihedral group, that's an R squared. And then the next one, you're gonna get R times five S, which again in the dihedral group, is gonna be a five R S. We're gonna write the elements in their normal form, then the next, we're not quite done yet, you're gonna get a two R squared, then you're gonna get a negative three R cubed, and then lastly, you're gonna get a five R squared S, like so. In which case, then we combine like terms. If there's any, do I see any constants? Just the two, so we're gonna slash that out to make sure we took care of it. Do we have any R's there? I see a negative three R, I see a two R, that looks like it, so that combines together to give me a negative R. So we'll mark those that we've then done. Do we have any R squares? We have a negative three R squared and a two R squared, so that combines to give me a negative R squared. So we're gonna remove those. We see that there's a negative three R cubed, like so. So remove it from consideration. And then let's see, what type of involutions do we have? We have a five S, we have a five R S, and we get a five R squared S. And so that would then be the product of these elements right here, okay? The product of these polynomial-like objects, or you might call it a group polynomial, determined by this dihedral group, okay? Well, that was if we do the product in one direction. As the dihedral group is non-commutative, it's non-Abelian, we anticipate if we were to do it a different way, we get a different product. So what I'm gonna do is I'm gonna turn this thing around, do the product in the other way, just to show you this illustration, for which case we're gonna get two minus three R plus five S, we times it by one plus R plus R squared, like so. For which when we multiply this thing out, we're gonna get two times one, which is two. We're gonna get two times R, which is a two R. We're gonna get two times R squared. Great, that's the first one. Then the next one, we're gonna end up with a negative three R times one, that's just a negative three R. Then we're gonna get a negative three R times R, which would be a negative three R squared. And then the last one, we get a negative three R cubed, like so. And then when we do with the S, we're gonna get five S times one, which is five S. Then the next one, we're gonna get a five S times R. I'm gonna come back to this one and do a little bit more detail. And then we get a five S times R squared in that situation. For which then with these last ones right here, you'll notice in the dihedral group, if you ever have an S times R, this is the same thing as R inverse S. And since we're working with the dihedral group D4, symmetries of the square, this is actually R cubed S. So we use the group multiplication to decide how to combine together these monomials. Likewise, if you have S R squared, because R squared, this would commute and become R to the negative two S. But as again, as we're working with the dihedral group D4, this would become R squared S. R squared is actually central in this example. So using the group multiplication, we then end up with these terms right here. This becomes a five R cubed S and this one becomes a five R squared S. For which then we would combine any like terms if we can in terms of a constant. We get a two in terms of an R. What do we get? We get an R, a two R and a negative three R so we get a negative R again. In terms of R squared, we have a two R squared and a negative three R squared so we get a negative R squared again. We now have a negative three R cubed. We're gonna get a five S. We're going to get a five R cubed S and we get a five R squared S which is not the same element we had from before. Of course, there's one critical difference that we see in the situation and that's gonna come between these elements right here. One product had a five RS, the other one had a five R cubed S and that's because again, as we multiply together these monomials, really you use the group multiplication to decide how things are gonna work, how you're gonna combine the indeterminates together, how you're gonna combine the variables. In a group ring, the variables so to speak are the elements of the group. They're not just these arbitrary symbols anymore like X and Y, they have some multiplication based upon the group structure. And so again, working with the dihedral group, if we had something like R squared plus R cubed, we wanna times that by say S plus RS, like so this would foil out in the usual way for which case you would get an R squared S, you would then get an R cubed S, you would get an R cubed S and then lastly you would just get an S because R to the fourth is one in the dihedral group for which then these things add together which you get a coefficient of two to R cubed S plus the last one of just an S. So that's how we do these things together. That is we multiply, we multiply and combine like terms the exact same way. The only thing that's different than usual polynomial arithmetic is that we use the group operation to decide how the multiplication of elements is gonna run across. Now I wanna further connect this idea of the group ring with polynomials which if you look at the group ring associated to the integers right here, right? Let's just take real coefficients and let's take the elements, the monomials to be integers themselves, right? And so as Z is typically written in added notation like zero and one and two and three, I need to make this into a multiplicative group. And so let's say that the integers are powers of some symbol X. So what I mean here is you could take X squared, X to the negative two, X to the negative one, one which is X to the zero, X which is X to the first, X squared, X cubed. So that is we can just look at all of the integer powers of X. And when we combine these things together if you take like X squared times X cubed, this just becomes X squared X to the two plus three power that is X to the fifth. In which case then you see a very natural isomorphism from the integers with their additive notation to this multiplicative representation of the integers, which basically just says you take the integer and you map it to X to the N. This is gonna be an isomorphism, but this way we can write the integers as a multiplicative group because we don't want two different meanings of addition in the same situation. Then in this setting, we see that the elements of this group ring themselves look like polynomials. I mean literal polynomials in the usual sense, not in this more abstract sense. The only difference of course is that we are allowing the possibility of negative exponents, which is something we do sometimes. Like for example, when you allow for potentially negative exponents with your polynomials, there's still integers, but negative integers, you get the idea of a Laurent polynomial. And so the Laurent polynomial ring, which is sometimes denoted as R to the bracket X, bracket inverse. So, because without just the R bracket X here means the polynomial ring with real coefficients and variable X, including the X and negative one who'd suggest, oh, I'm allowing for X and negative powers of X as well. This is called the Laurent polynomial ring. And the Laurent polynomial ring is just a special case of group rings because it is in fact the integer group ring. And so we can create rings using a coefficient ring R and a group G, where again the multiplication of our monomials is determined by the multiplication of the group. Group rings are actually a very fascinating topic and a very active topic of research in abstract algebra.