 In this video, we define the notion of an ideal for a ring, and we had mentioned how an ideal is basically the ring theory version of a normal subgroup. They're kernels of ring homomorphisms. You use them to make quotient rings. And so I want to then add to this theory of ideals we've talked about previously. So it should be mentioned that if you have any ring R, right, every ring has two automatic, you know, we could say natural ideals that, again, every ring is going to have this. And this is an analogous statement to what we did with normal subgroups for a group. So one, you have the so-called zero ideal, all right. And so this would just, of course, be the set that only contains the zero element of the ring. Notice that to be an ideal, we have to be closed under addition. Now, of course, zero plus zero is equal to zero. So that's pretty straightforward. Now, we also have to be closed under the ideal multiplication, as in, if I take any element of the ring and times it by any element of the ideal, it must be in the ideal. But that's pretty easy here from properties of rings. If you take any element in times of by zero, and it doesn't matter which sides you are, even in a non-commutative ring, this is always equal to zero. Zero, because it's the additive identity and because of the distributive laws, inherits this absorption property that anything times by zero gives you zero. Zero is a dominant ring element. And as such, this, the ideal that only contains zero, the zero ideal, is always an ideal for any ring whatsoever. Now, as an abusive notation, the zero ideal is often denoted as just the symbol zero, because these little subsets, subsets, symbols can be tedious to type in, right, sometimes. So we often just denote the singleton by the one element zero itself. So hopefully that's not too confusing with people. We do a similar thing for group theory. We often say, like, the trivial group is just the element one. So we say things like that. Really, we're saying, oh, the trivial group is the, the, the subset that only contains the identity. But we drop the set notation, just call the element the name of the set. We'll do the same thing with the zero ideal. All right. The other, the other ideal that's always guaranteed is, of course, we can take the entire ring itself. So every ring is an ideal of itself. And I should probably denote this as well. The zero ideal is an ideal there. The ring itself is an ideal in a trivial manner, of course. Because again, it's going to be closed under addition because it's a ring. It'll also be closed under multiplication. If I take any element from the ring and I times it by any element of the ring, that'll give you an element of the ring. So the ring itself is an ideal. Now we have these two guaranteed ones. And we saw an analogy for this, of course, with with natural normal subgroups inside of groups, right? The trivial group, trivial subgroup is always normal. The whole group itself is always normal. And then as we studied simple groups, those were groups who had no other normal subgroups other than the trivial one and the whole group itself. So rings have sort of this analogy here that we have the, the smallest possible ideal, the largest possible ideal. What about those in the middle? Now, it turns out that we will be interested in the ring theory equivalent of a simple group, right? What are rings that have no non-trivial proper ideals? Now the study of simple rings is a lot easier topic than, of course, it is for simple groups, I should say, because the distributive law means there's some type of interaction between addition and multiplication. And that does force a lot. That forces a lot of conditions here. And so while I'm not going to give the conversation about all simple rings, it is still one that we could do. One could talk about, of course, at a commutative simple ring. They're a lot easier to discuss. And it turns out that the presence of units make, make, they make ideals difficult to form. And let me explain this in the following way. So assume for the sake of this following conversation that we have a ring with unity. So there's some, there's a multiplicative identity here. You can talk about simple rings without unity. It gets a little bit more complicated. It's not too complicated, but it's a lot easier when you have unity in the conversation. Commutivity or not, don't care. Imagine you have some type of unity inside of this ring. Okay. So what I want you to do is consider a unit. So U is an element of our ring and it is a unit. Okay. What does that mean? Well, that means there is some multiplicative inverse that this is equal to one. I want you to imagine that we have an ideal, some ideal inside of our ring here, but say that a unit belongs to the ideal. Okay. So in that situation, it must be true. It must be true that if you take any element of the ring and you times it by an element of the ideal, then it's inside the ideal. In particular, okay, if U is in there, that means that U times U inverse is in there, because even if U inverse is in the ideal, since U is in the ideal, anything times U is in the ideal, so U times its inverse is in the ideal, that gives you one. One is in the ideal. So if I take any element of the ring R, and that I could always factor R as R times one, since one is in the ideal, that means R times one is inside the ideal. And so we get that every element R is inside the ideal. This then shows that R is a subset of I. Clearly by construction, I is a subset of R, so we would then have to then conclude that the ideal equals R. So the reason I bring this up is it's very important to remember that if an ideal contains a unit, it is the whole ring. And so if you're interested in what is a simple ring, what is a, why can't I spell simple here? Sorry about that. If you want a simple ring with unity, this is just going to be a skew field, okay? Because if every element is a unit, then every ideal is going to be the whole ring, except for the zero ring, of course, because in a skew field, everything's a unit except for zero. But does it go the other way around, right? So we know a skew field will be simple because it has no ideals for this observation we made here. But if every ideal is basically, if there's no non-trivial proper ideals, why does that make it a skew field? Why does that make everything a unit? Well, that's actually what I want to talk about right now. So you might have noticed there's this text to the left I haven't said much about yet. So I wanted to find the idea of a principal ideal, which is essentially like a cyclic subgroup for a group, if we could make an analogy there. So imagine we have a ring, and I'm not going to make any assumptions about whether it's unity or commutative or anything like that. We just have a ring, so the multiplication is associative. Let A be an element of the ring. Then we can define the following set, which we're going to do it as bracket A bracket. So similar to notation we use for group theory here, and this is going to be the span of all elements of the form R times A times S, where R and S are potentially any element of the ring R. And so what do we mean by span right here? This is a linear algebraic term. The idea is we're going to take the sum, the sum of elements that look like this, R plus S. So a typical element of this set would be something like some R1A, S1 plus R2A, S2, all the way up to some RNA, SN. So we have a bunch of things that look like this. So we take all the possible sums of those things. That's what we, of course, we mean for this span of things like that. Again, the R's and S's are completely arbitrary. This is referred to as the principal ideal generated by A. Why is this going to be an ideal? There is an argument there and it's not too difficult of an argument. Well, after all, if you take a sum of things like this and you take another sum of things like that, that's just a bigger sum. But that's the idea is if you take a sum of things that look like RAS and you add that to another sum, you have a bigger sum of things that look like RAS. So it's closed under addition in a trivial manner. Spans are always going to do that because by construction, they're all the possible sums of things that follow a certain profile. So if you add more things to it, it's still going to be a sum. Multiplication, why do we have stronger multiplication? Well, if you take an element like this and you times it by some element R, well, the distributive property holds and you're also associative. So this would look like RR1 times AS1 and this would proceed all the way down to RRN, ASN. So this still has the same form. This belongs to the ideal. I did multiplication on the left of multiplication on the right by a ring element would be very similar. So the principal ideal generated by an element A is in fact an ideal, deserving of the name. Now, if R is a commutative ring, then the definition of a principal ideal can be dramatically simplified. We define this just to be, well, just the set of things that look like A times R. Why can we get away with that? Well, again, multiplication is always associative and now we're assuming it to be commutative. So if you had something like RAS, this then factors, well, not factors, but you can write this as ARS. So if there were something on the front here, you could always move it to the back and put those things together. And then why is it closed under sums? Well, if you had something like AR plus AS by the distributive property, this is A times R plus S, where R plus S is just something in the ring. And so for commutative rings, these principal ideals are a lot easier to describe. One sort of interesting hiccup that I do have to mention when it comes to principal ideals is the following observation. If you have a ring without unity, so there is no multiplicative identity, then it's actually possible that the principal ideal generated by A doesn't even contain A itself. A might not be in there because in this setting, let's say you have a ring with unity, then we'll also just assume it's commutative and just make life easier for us. So then in that situation, A can be factors A times one, and so that then belongs to this principal ideal because you have unity. But if you don't have unity, you don't necessarily have a guarantee that there's any product of A that involves A itself. So it's kind of a strange phenomenon that A wouldn't even belong to its own principal ideal. But again, without unity, that could be the case. So you have to be very, very cautious in that situation. We're not going to deal with very many rings without unity in this course, but I should mention that ideals themselves, since they generally aren't going to contain units, they typically won't have ideals. They won't have, since the ideals don't have units in them, they might not have an identity in them, a unity. So ideals are often sub-rings without unity. So it's not that we don't ever talk about them really, but just be cautious. It's sort of a weird anomaly that can happen. But of course, it needs to be said that if we do have a ring with unity, then a principal ideal, the principal ideal of general A is equal to the ring R, if and only if A is a unit. So there's some value A times R that equals to one. And so this then connects to what we were talking about earlier about skew rings, that a ring with unity, that simple ring with unity, ring with unity, I guess we usually do it the other way around, ring with unity. This is exactly what we mean by skew field. And so in particular, a commutative, a simple commutative ring with unity is a field because look at a principal ideal. So if you take the principal ideal generated by zero, you're just going to get back the zero ideal. But if you take an ideal generated by something that's not zero, you're not going to get the zero ideal. You're going to get something bigger. Now, if you're a simple ring, that means you would have to be everything. Okay, but a principal ideal with a ring of unity is only equal to everything if it's a unit. Because if your ring has unity, that means one is inside your principal ideal. And if one's inside your principal ideal, that would mean there's some ring elements so that a times that ring element r is equal to one. That's exactly what a unit is. So simple rings with unity are exactly just skew fields. If you take away unity, like I said, simple rings get a little bit more complicated, but not too much. But this is an important observation to make about principal ideals, what they do with units, and this idea of simple rings, not having ideals. I should also mention that in ring theory, an alternative notation that's very common to denote principal ideals, instead of using angle brackets, you actually use parentheses. And although this is really common notation, I mean, it's used all over the place, you should be aware of it. In our lecture series, we're going to instead use these angle brackets. These parentheses, the parentheses a can sometimes mean other things. So there is sometimes some ambiguity that happens. And I want to try to avoid that in this lecture series, not try to confuse anyone with it. And so I'm going to use the angle brackets. I also want to paint a parallel between the cyclic subgroups we talked about in group theory with these principal ideals we're talking about now. Alright, so in this video, I want to talk about the ideals over the ring of integers. And we're actually going to prove the very simple proposition that all ideals in the ring of integers are actually principal ideals. This is going to be a, this is a foreshadow into an idea we're going to talk about later on about a principal ideal domain. And this proof is essentially that the integers form a principal ideal domain. So okay, so the first thing is like I said before, the zero ideal is itself a principal ideal because it's the ideal generated by zero. So what we have to do is prove that any non-zero ideal is in fact principal in the case of the ring of integers. So imagine we have a non-trivial ideal inside the integer ring. Alright, so one thing that's important to mention is that if that ideal contains any integer non-zero letter m, right, I can assume that it's gonna, I can assume m is a positive integer, right? Because if m was negative, then its additive inverse is also in there because it's closed under addition. If we look at just the additive structure, an ideal is a subgroup, an additive subgroup. So if you have m, it has its additive inverse as well. So you're gonna have a plus or minus m. So one of those is going to be a positive integer, whether it's m or negative, either one. So without the loss of generality, I can select a positive integer if I want to inside of this ideal, okay? And so because I does contain positive integers, by the well ordering principle, the ideal has to contain a smallest positive element. Because after all, if we take, if we take the ideal i, it intersected with the set of natural numbers, that's a subset of the natural numbers, and the well ordering principle applies there. It has a smallest element. We're gonna call that smallest positive element n, and it lives inside the ring, the ring, the ideal i right there, which of course, you could also look at negative elements, but it'll be easier just to look at the positive ones here. Now since it's an ideal, if I take anything times n, so n times an integer will belong to the ideal, that's the ideal multiplication closure there. And so this is an important observation here that the principle ideal generated by n is always a subset of i here. And this is actually a principle and truth, truth for basically any ring, right? Well any ring with unity, right? So if you have a ring with unity here, then if you have an element a that's inside of an ideal, then you're gonna have that the principle ideal generated by a is going to be a sub ideal of that ideal. So that's an important observation to make right here. So this is just a statement because i is an ideal and you have an element n that's inside the ideal, it's the principle ideal generated by n will be a subset. Now our goal is of course to prove that the other direction holds, that the ideal is none other than this principle ideal. And so this is where the this is where properties of integers are going to come into play, kind of like the well-ordered principle we just mentioned a moment ago. So imagine a is some other positive element that lives inside of the ideal. By the division algorithm, there exist positive integers q and r such that a equals nq plus r and we can assume that r is a non-negative number less than n. All right, could be zero possibly positive but it's less than n, all right? So of course, since a is inside the ideal and we already mentioned that nq is inside the ideal, notice that a minus nq which is equal to r, this lives inside of the ideal. But n was the minimal element of i, the smallest positive element of i. So we know that a, excuse me, if r is not zero, r is a positive element and it has to be smaller than n and it's contained inside i. So that would be a contradiction unless of course n is equal to zero, excuse me, r is equal to zero. So of course if r equals zero, then this actually implies that a equals nq and q is a typical element of the principle ideal generated by n, excuse me. In which case then we get a which equals nq belongs to the principle ideal for which of course the integers, every principle ideal, you know, take it generated by n, that's just the set nz. So every positive element of an ideal belongs to this principle ideal, right? Now if you took a negative element, well then its additive inverse is in there as well, its additive inverse belongs to this, which then since the the principle ideal generated by n is itself a principle ideal, I mean it's an ideal, its additive inverse will be in there as well. So every element of i belongs to the principle ideal generated by n, its smallest positive element. And so this then gives us the other direction that every ideal is in fact equal to a principle ideal. And this will be a very crucial observation that we use in the future as we continue to develop our theory of rings and then ultimately with domains.