 A domain is a ring with unity equipped with the zero product axiom. It turns out that that is equivalent to taking on the cancellation axiom. That is to say, R is a ring with unity, then R is a domain if and only if it satisfies the cancellation axioms, both left and right cancellation. So in a ring, if you have the zero product axiom, you have the cancellation axiom. And if you have the cancellation axiom, you have the zero product axiom for rings with unity. So that's what we're going to prove right now. So and just a reminder, the cancellation axiom here is on the screen. We can either right cancel or we can left cancel. All right. So let's first assume we have a domain. So therefore we have the zero product property. And I want you to consider an equation for which cancellation would apply. AB equals AC. We want to prove that B equals C. Now, because this is a ring, we can move AC to the other side by subtraction. AB minus AC is equal to zero. There's a common divisor of A. So we can factor it out and we end up with A times B minus C is equal to zero. Okay. And so now we have a product of things equal to zero. Now, can we cancel A? Now, what they have to point out here is with regard to the cancellation axiom, we are only supposing we can cancel elements which are non-zero. Okay. You can't cancel a dominant element. And therefore the cancellation axiom supposes that you can cancel any non-dominant element, any non-zero element. So since A is non-zero, since we're in a domain, that means the other factor has to be zero. B minus C is equal to zero. But if you add C to both sides, we then end up with B equals C. And then we've proved here the left cancellation property. Right cancellation has proven identical to that. No big deal. So now let's suppose cancellation holds in this ring that the only number we can't cancel is zero. Everything else is able to cancel. Now consider a product of two numbers that are equal to zero. All right. Well, if the first number is equal to zero, then the zero product property doesn't apply in that situation. It only cares when you have a non-zero product. So okay, so that's we're done. If the first factor is zero, so suppose A is not zero. Well, if AB equals zero, we also know that A zero equals zero because zero is dominant. You can cancel A from both sides and we end up with B equals zero. Therefore, we've proven the zero product property that the only product of numbers that's equal to zero is one of the factors had to already be zero, either A was zero or B was zero. So for rings, having cancellation is the same thing as having the zero product property. A domain, which by definition takes on the zero product property, you also have cancellation. And so this is a very important thing about domains. Commutivity was never assumed. This is also true for integral domains, but we don't need it to be commutative. For an integral domain, you have cancellation. And this is something that's like a pseudo field that while you don't have inverses, you can still solve equations using cancellation, using the zero product property. And so an integral domain is a ring that's very close to the type of things we do in a standard, you know, non-abstract algebra class.