 So the theory of rings is very similar to the theory of groups. We've introduced the axiomatic definition of rings in our previous video and gave you some examples. And then if we progress to the same story that we went through groups much faster though, a very natural thing to talk about next are rings inside of rings, that is so-called sub rings. Let in, excuse me, let R be a ring and let S be a subset of R. Now, we say that S is a sub ring of R denoted S is less than or equal to R just like we do with groups. If S is an additive subgroup of R and it's closed under multiplication, that is a sub ring is gonna be a ring inside of another ring using the same operations of addition and multiplication. That is if we just restrict addition and we restrict multiplication, we get a ring structure still. We say that R is a, well, if R is a ring with unity, we say that S is a sub ring with unity if S is a sub ring and it contains the unity of R. We have to be very careful in this definition here. So when we talk about a sub ring with unity, it's a sub ring that contains the unity of the larger ring. A sub ring with unity does not mean that it's a sub ring that has unity. It has to be the same unity. It's very possible in fact that you can take a sub ring of a ring which has a unity which is different from the unity of the larger ring. That could be, for example, because the larger ring has no unity to it whatsoever. Like if we take, for example, the ring of matrices of the following form, let's say let's take matrices AB00, something like this where A and B are just real numbers. You can prove that this set of matrices is a ring with respect to matrix addition and matrix multiplication in the usual sense, but it's without unity. There is no matrix which acts like unity for this ring. But on the other hand, we can take the subset S which consists of let's say A00, like so, or A is a real number. So this would be a one-dimensional sub ring in this situation for which this one does have, this does have a unity. We could say one or sometimes they call it U for unity here. This would be the element one, zero, zero, zero. You could show that this acts like a unity on the sub ring S, but that's not a unity for the whole ring, right? And so that's kind of a curious observation. We have a sub ring which has unity, but it's not the unity of R. So we wouldn't call it a sub ring with unity. It's a sub ring, but it does have unity. I know the language might seem a little bit weird here, but this is an important distinction. What's even more bizarre is that you can have a sub ring of a ring with unity and the sub ring is a ring with unity, but because it's a different unity, that doesn't count as a sub ring with unity. So for example, consider the ring R cross R. So we could define the direct product of rings analogous to how we define the direct product for groups for which as a set, we'll just take the Cartesian product of the two rings and play here and we define addition and multiplication component-wise. That is, we just add together the first component to add together the second components. We multiply together the first components. You can multiply together the second components, all right? And so this, you can argue that the direct product of two rings forms a ring. Consider the sub ring S defined to be all of those ordered pairs X comma zero where X is a real number, like so. You can show that S is a sub ring of S, okay? In which case you'll then see that R has a unity. I'll let you think about what that unity would be. S will have a unity and S is a sub ring, but the unity in S is not the unity in R. In fact, S doesn't contain the unity of R, it has a different unity. And so it's very important when you think of rings, sub rings with unity or not, to be a proper sub ring with unity, you have to have the same unity as before. So just kind of make some comments about that. It's kind of a weird thing because the restriction of the set does not guarantee that you contain the unity and there's no access, don't guarantee that if you have unity, it's the same unity. That's something very different we saw when it came to groups. If you had H as a subgroup of G, for example, and if this is a subgroup H, then it's a group inside of a group in which case it has an identity. So how do we know that the identity of H is the same as the identity of G? Well, the short answer was cancellation. Because we have cancellation inside of H, you could then infer that the identity of H has to be the same identity of G. In general rings, we don't have cancellation of multiplication, that's a topic we might talk about some other time. And because of that, we can't guarantee that the unity of the sub ring is the same unity as the mother ring there. Now, some examples of sub rings that we're familiar with, like we're familiar with the field of complex numbers. The field of real numbers is a commutative sub ring with unity. The field of rational numbers is a commutative sub ring with unity as well. And being a sub ring is a transitive relationship here that since Q is a sub ring of R and R is a sub ring of C, then Q is a sub ring of C, right? And they all have the same unity. Likewise, the integers form a sub ring of Q, which is in a sub ring of R and C. And this is a sub ring with unity. All of these guys have the same unity. And of course, if you take the integers, you could take any, you take your favorite integer Z, then NZ, this would be all the multiples of N. This forms a sub ring of Z. It's a commutative sub ring, but these are not sub rings with unity because NZ itself doesn't contain any, doesn't contain any, it doesn't contain unity.