 In this video, I wanted to find the notion of a maximal ideal. So let R be a ring. We say that M is a maximal ideal if, well, first of all, it's gotta be an ideal. Second, it can't be the whole ring. We've seen previously that the whole ring itself is an ideal, the improper ideal. So a maximal ideal by definition must be a proper ideal. But the term maximal here suggests that also there's no proper ideal I that properly contains M. In other words, if M is a maximal ideal and I is any other ideal that's sandwiched between the maximal ideal and the whole ring, then it must be that this ideal I is either M or it's R. There's nothing in the middle. In other words, there's no ideal between the maximal ideal and the whole ring. That's what we mean by a maximal ideal. And from this, we get a very, very simple and very important property about maximal ideals. So an ideal M of a commutative ring with unity R is maximal if and only if R mod M is a field. Okay, this is essentially a consequence of the correspondence theorem. If M is a maximal ideal, then by the correspondence theorem, R mod M has no proper ideals. Because since M is maximal, there's no ideals between M and R. By the correspondence theorem, every ideal of R mod N can be lifted to an ideal that contains M. Well, by maximality, any ideal that contains M is either M itself or R. Now, if you mod out by M, then M becomes the zero ideal. And if you, well, then the whole ring, of course, R itself you've mod out by M, that's the whole ring. So there's no other ideals. So because we have no other ideals, there's just the two ideals, the zero one and the whole one, that makes it into, as we talked about before, that makes it into a skew field. Since we're assuming it's commutative, that of course we make it a commutative skew field, which of course is just a field, right? Or again, if you missed that argument from before, we'll just say it again. Take an element R that lives inside the ring, but outside of the maximal ideal. Then if you look at the principal ideal generated by that element R plus M, because there are no proper non-trivial ideals, this principal ideal has to generate the whole ring. And if a principal ideal generates the whole ring, then it has to be a unit. Every element of a commutative ring is a unit that makes it into a field. So that direction is essentially just the correspondence theorem. Since you kill off a maximal ideal, there's no ideals left. And so every principal ideal is the whole ring. You then get those every elements of unit. And conversely is basically the same thing, right? The reverse direction is gonna be the same thing. If you imagine the R mod M as a field, then it has only two ideals, which are gonna be zero and RM. That's because every element that's not zero and RM is a unit because we're assuming it's a field. And a unit, the principal ideal generated by unit always forms the whole ring. So then my correspondence, since RM has only two ideals, there can only be two ideals of R that contain M. One of them is M, one of them is R, that's it. So if we count it for all of them, therefore this is a maximal ideal. So in this theorem, we assume that R was a community of ring with unity, but I want you to be aware that if we drop the word commutative, this would just mean that an ideal is maximal in a ring with unity, if and only if R mod M is a skew field, the commutivity there doesn't really, it's not necessary. You can also even drop the word with unity, in which case we can talk about a maximal ideal, in which case in that situation, what you get is M is an ideal of, is a maximal ideal of R, if and only if R mod M is a so-called simple ring, meaning it doesn't have any ideals other than zero and the whole thing. Which again, these general simple rings we've talked about before in this lecture series, we're not gonna say much more about them than do exist, but we won't say much more about them. The reason we chose the assumptions we did commutative ring with unity, it's because that's really the setting that we wanna be in. In this class, we're mostly focusing on factorization theory inside of integral domains, which are rings which are commutative with unity. So in that setting, we wanna study maximal ideals and so R mod M exactly will give you a field. Let's consider a short example before we do in this video on maximal ideals. If you take the ring of integers, which of course is a commutative ring with unity, if you mod out by the ideal PZ, which we have studied before that in the ring of integers, every ideal is a principal ideal and every ideal is just has the form P times Z, right? This will be congruent, this will be isomorphic to the ring, the finite ring Z mod P. And this is gonna be a field if and only if P is a prime number. That's something we've studied before. In which case then by the previous theorem, if this is a field, that means this is a prime, excuse me, this is a maximal ideal. And so we see that the maximal ideals of the integer ring are exactly those generated by prime elements. So in the ring of integers, the maximal ideals are those generated by a prime element. And that's then gonna lead naturally into the topic that we'll talk about in our next video.