 Imagine we have a commutative ring R and let P be a proper ideal of this commutative ring. We say that P is a prime ideal if whenever the product AB is found inside of the prime ideal P, then it must have been true that the factor A was in the prime ideal or B was inside of it. So anytime a product is inside the ideal, one of the factors, not necessarily both, but at least one of the factors was inside of the prime ideal. And so I want you to try to understand what's happening here. This idea of a prime ideal is trying to generalize the notion of a prime number because prime numbers we have defined for of course the ring of integers. We want to sort of generalize this beyond that because if you take a prime number, we can look at the principle ideal generated by that. And this prime ideal is in fact, excuse me, this ideal principle ideal generated by a prime number is in fact going to be a prime ideal that we're going to see in just a little bit. And so this idea of prime ideals is really has to do with Euclid's lemma. So Euclid proved for prime numbers, these rational prime numbers, right? The ones inside the field of rational numbers. Just usual integers is what I'm talking about right now. You could prove the property that if a prime number divides a product A times B, then that implies that P divides A or we get that P divides B. So one of those two things happens. And so this idea of prime ideals is trying to capture that. We call it ideal prime if it satisfies Euclid's lemma right here. If it has the same property that Euclid proved for prime numbers. And so what does this have to do with ideals? What does divisibility have to do with ideals? When you move to a principle ideal, it's exactly that. So imagine we have a principle ideal generated by some element P. And imagine that AB is inside of that, okay? Well, if AB is inside of that, that means there exists some element R inside the ring, such that PR is equal to AB, okay? So that then gives us that P divides the product AB, okay? So if AB, if a number, if a number, this is in general, if a number is inside of someone's principle ideal, then the generator of that principle ideal divides that element, okay? So if AB is in there, you get that. So then if Euclid's lemma applied, right? If this element P is quote unquote prime, then that would mean that P divides A or P divides B. Now if P divided A, that meant, oh, you have some P, some PS is equal to A or something like that. In which case then this would belong to the principle ideal generated by P. So my point is this idea of divisibility is equivalent to containment when you talk about principle ideals. Not every ideal is necessary principle, so this is the appropriate generalization. An ideal is prime if it has this Euclid lemma-like property, okay? And so it turns out that similar to our, well, I should say that prime ideals have a property similar to what we talked about previously with maximal ideals. An ideal of the commutative ring with unity R is a prime ideal if and only if R mod P is an integral domain. So remember, integral domain would be a commutative ring with unity with no proper divisors of zero, or you could also say it satisfies the cancellation property. And so modding out by a prime ideal forms you an integral domain and vice versa. That is, if your quotient is an integral domain, you mod it out by a prime ideal. So assume that P, let's first do the first direction, right? Let's assume P is a prime ideal and let's consider a product. So imagine we have a product of two cosets here, A plus P times B plus P and imagine that equals zero plus P, aka P. So this is the zero element of the ring R mod P. So what we're saying here is we have a factorization of zero. If this was in fact an integral domain, let's show that one of the cosets A plus P or B plus P was already the zero element, aka it was already P. All right. Now these elements A and B are elements of the ring. And so if we summarize this, I mean, if you take A plus P and you times it by B plus P, this, the way that coset multiplication works is it becomes A B plus P and this is supposed to equal P. Well, when this happens with cosets, this actually means that A B is an element of the prime ideal. But by assumption, since it's the prime ideal, this means that either A belongs to the ideal or B belongs to the ideal. Without the lots of general analysis to say it's A, it's commutable multiplication. It doesn't matter the order here. So we're assuming that A plus P equals P. This is the zero element of R mod P. And so when you have this product right here, it's like, oh, here's the zero element. We have a product equal to zero. Well, one of the factors is equal to zero. So this then shows us that the ring R mod P has no proper divisors of zero. Since it's commutative, since it has unity, that then implies that it's an integral domain. And I will leave it as an exercise to the viewer here to prove the other direction. Assume that R mod P is an integral domain and then prove that P is a prime ideal. The calculation, the proof is very, very similar to what we just saw here. But I want to make a connection to what we did in a previous video here. So we've just proven that R mod P is a domain if and only if P is a prime ideal. So we have that R mod P is an integral domain if and only if P is a prime ideal. Now, in a previous video, when we talked about maximal ideals, we learned that R mod P, where P is still the ideal in question here, this is a field if and only if R, excuse me, P, I should say P is a maximal ideal. But it's also important to remember that the set of fields, like the category of fields here sits inside the category of integral domains. And particularly every field is an integral domain because you have units, you have no zero divisors. So since every field is an integral domain, you then get the following argument. Every maximal ideal is a prime ideal because you start off with a maximal ideal. So M is a maximal ideal. So that's step one. So then you mod out by M, you get a field. You then infer that, oh, because it's a field, it's going to be an integral domain. And then because it's an integral domain, that means M is prime. So in a commutative ring with unity, every maximal ideal is prime. That'll be a very useful property as we go forward and study factorizations and domains, which is going to involve statements about their ideals.