 In previous videos of this lecture series, we introduced the idea of fields and skew fields, which these are rings for which every non-zero element is invertible. It has a multiplicative inverse or a so-called reciprocal. Now, not every ring we've studied is a field or even a skew field, right? Like the integers, right? It's the only invertible elements there are gonna be plus or minus one. Most elements are not invertible. If you take a polynomial ring, even if the coefficient ring is a field, the polynomial ring itself is not gonna be a field or skew field. Group rings also have that characteristic that even if the coefficients are a field, right? Group elements themselves, every group element is invertible, but when you make a ring out of it, you're gonna create lots of elements which are not invertible. Same thing to be also said for matrix rings. So when we study rings, we're very interested in when is an element invertible or not? We're interested so much that we give it a name. If R is a ring with unity, which if we don't have unity, there's no reason to talk about reciprocals. If R is a ring with unity and we take some element little R inside of R, we say that little R is a unit if there exists some multiplicative inverse. That is the ring contains some multiplicative inverse of R. We call it R to the negative one right there. And so we're very interested in when we study rings in units. What elements are units? How many units does the ring have? What are the units? How can we classify the units? So the set of all units inside of a ring with unity is often denoted R star. Sometimes it's denoted U of R. So the set of units right there. Now I wanna mention that the set of units inside of a ring forms a group which is necessarily a multiplicative group. Why is it multiplicative? Okay, well that's actually, there's a nice little argument there which clearly one is a unit. One belongs to R star. And this just comes from the fact that if you take one times one that's equal to one so it has an inverse itself, right? The inverse of one is itself one. I also wanna show you that if you're a unit then you have a multiplicative inverse, duh. So if R is inside of R star that means that R times R inverse equals one which equals R inverse R. In particular it has an inverse R star necessarily. Well, R negative one, excuse me but R negative one likewise has an inverse that's just gonna equal R because of the uniqueness of inverses. That's gonna come down from the associativity axiom here. And so if R is in R star then R inverse is gonna be in there as well. So we get that this set of units is closed under multiplicative inverses. Closure is also pretty easy. If you have R and S which are inside of R star right here then we know that there's some R inverse, there's some S inverse. And so what I'm gonna do is I'm gonna take RS and I'm gonna multiply it by S inverse R inverse, right? Thinking of the shoe sock principle you have to reverse the order when you do the inverses. Then by associativity this will equal R R inverse which will equal the identity and therefore S inverse R inverse is equal to RS inverse. And this is a two-sided inverse. If I go from the other side you'll notice that S inverse R inverse times RS the RS cancel and then the S is cancel and you get one again. So we get that RS will be inside of R star. So this is in fact a group, it's called the group of units and it's a very important question as we studied ring theory what is the group of units for that ring? In fact, a group, excuse me, a ring is a skew field if and only if the group of units is everything that's non-zero. So let's look at a few examples of the units of some rings some of which I've already alluded to. So if we take the ring of integers this is not a field like I mentioned earlier the group of units is gonna be plus or minus one the only integers which are invertible will be plus one the unity and negative one it's additive inverse. You can also argue that if an element is a unit then it's additive inverse is likewise a unit if say U belongs to R star then that means there's some U times U inverse that equals one. Well, what happens if we take say negative U? Well, the inverse of negative U is gonna equal that is the reciprocal of negative U is actually negative the reciprocal of U there and we can see that very easily from properties we've proven previously notice if I take negative U and you multiply that by negative U inverse so that is I'm taking the additive inverse of the reciprocal of U based upon properties of rings we've seen already then this will give you U times U inverse that is a double negative as a positive and that gives you one there. So inverses that are units inside of ring their additive inverses are likewise units, okay? So in that case for a ring with unity you always get plus or minus one as units but then it turns out in the integers there's no other units you have sort of these two guaranteed units but no other ones, right? The integers don't have a lot of units. What about Zn for example if we take the ring of modular arithmetic n well we've talked about the group of units there Zn star we actually use the notation Zn star sort of forecasting foreshadowing this group of units we had before and so we know for Zn star this consists of all integers which are co-prime to n and we've studied this group extensively so if n has very few divisors it'll have very many there'll be a lot of units inside of the rings Zn in fact like we've seen previously if n is a prime then Zn is actually a field so everything's a unit except for negative one. How about the matrix ring m that is let's take all the n by n matrices over the coefficient the scalar ring r in that situation well if we're looking for the units then as we're looking for those n by n matrices which have a multiplicative inverse that's what we usually call a non-singular matrix and so the group of units for a matrix ring is actually what we call the general linear group so we've studied this group a lot in this series as well for which we've talked about the general linear group with real and complex coefficients that's the unit group for the ring of real matrices or complex matrices and this can be generalized to any field or any ring in fact we can define the matrix ring with any coefficient ring in which case then we can also talk about the general linear group over any other ring this would be the group of invertible matrices so we could talk about the general linear group of n by n integer matrices right here which it could be there might not be a lot right you might have the identity matrix you might have negative one times the identity matrix but there's not gonna necessarily there might not be a lot now interesting there are a lot of integer matrices whose inverses themselves are integer matrices for example if the determinant of an integer matrix is plus or minus one it's inverse will likewise be an integer matrix so this group does contain all integer matrices of plus or minus whose determinant is plus or minus one but of course you can get some other things in there as well okay when it comes to polynomial rings the unit group is actually very simple if you have the polynomial ring Rx then the units are just the units of R that is R bracket X star is just R star and what I mean by that is the only polynomials which are invertible that are units are gonna be constant polynomials which themselves are invertible and that's from the following argument here that if you take the degree of a polynomial if you take the degree of F times G right you have two polynomials you multiply them together the degree of their product is going to be the degree of F plus the degree of G and the main reason behind that is when you multiply together polynomials if you take the leading term of one let's say that's powers M and you times it by the leading term of the other that's N then you add together their degrees M plus N and this never reduces down for standard polynomial multiplication right here and so the degree always gets bigger when you multiply two polynomials together it can never get smaller now it could stay the same because you can have degree zero polynomials for which a polynomial its degree is zero if and only if F of X here is a constant like say some C right here in fact the unity of a polynomial ring is gonna be the constant polynomial F of X equals one so the only way which then its degree is zero right so the only way that the product of two polynomials degree can equal zero is if they were already constant polynomials you have to have a constant times a constant for which if you're a constant then you're essentially just an element of the coefficient ring in which case you didn't have to be a unit of that so okay so some of these rings their unit groups are groups we already know very well like these ones some of them are also easy to describe so polynomials don't offer anything new so the last example I wanna talk about with respect to units is a little bit more complicated it's the idea of units inside of a group ring okay so let's say we have some group G some ring R then the calculation of units is a lot more difficult now there are some obvious situations like if I take the units of the coefficient ring that'll be inside the unit group much in the same manner here although equality we would not expect we also have that the group itself would be inside of the unit group of the group ring okay and the reason for that is if you have a group element the group element can be viewed as an element of the group ring right and each of these are invertible because if you multiply it by its inverse which is also an element of the group ring you get the identity back the multiplicative identity so the units inside of the coefficient ring will be units inside of the group ring and then the elements of the group themselves are and so these two classes of units are often referred to as the trivial units inside of a group ring we predict those ones will be there and so what many people who study group rings are concerned about is that are there non-trivial units that is is there some combination of group elements so that you get a unit let me give you such an example so let's take the group ring over the symmetric group S3 and to keep things simple we'll just have integer coefficients it turns out the coefficient ring matters a lot on what you can do to make a unit or not but I'll give you one with integer coefficients okay so let's introduce the element mu it's going to be one plus the three cycle one two three minus the three cycle one three two plus the two cycle one three minus the two cycle two three now as a group ring these elements we can't add or subtract these elements together these are like different terms it's like in a previous algebra class if you take X plus Y can you simplify that no the formally that's the sum they can't be we can't combine any like terms there I claim that the inverse of mu is going to be one minus one two three plus one three two minus one three and plus two three which you'll notice basically I changed the sign of all of the non-identity none of all the non-constant terms in that combination I don't claim that'll work in general but it kind of feels like it's a conjugate in like a complex conjugate sense or a quaternion conjugate sense again I don't claim that's going to work in general there's a little bit more going on behind the scenes here don't look at the man behind the curtain so if we actually work out the details of mu times mu inverse I have them on the screen right here feel free to pause it and work out the or better yet work out the details yourself but as you work through the products inside of this I mean there's a lot going on here I mean this has five terms in it this has five terms in it so as you multiply all of these things out it can get a little bit messy it can I mean there's like 25 possible products you have to compute but what you're going to see is that everything is going to cancel out when you're done you do have an identity you have a number one right here that's the only thing that's going to be left around notice I've organized the data in such a way that you see that the one two three one two three cancels one three two one three two cancels one two cancels one three cancels two three cancels two three cancels one three cancels one two cancels you see all this cancellation boom and boom everything's going to cancel out except for a one so this is in fact an authentic unit of the group ring and it's going to be a non-trivial unit and so that's kind of where I want to end this video here I just mentioned that the study of non-trivial units in the group ring is a much much more complicated problem the study of units is a very very active field of research in modern abstract algebra because just from a purely mathematical point of view it's a very interesting question very intriguing but also there are so many practical applications believe it or not finding units inside of a group ring is very related to the coding theory problems we developed earlier in this lecture series so developing a more sophisticated more effective error detecting error correcting codes has a lot to do with recognizing units inside of a group ring.