[MUSIC] It is clear why it has for this [UNKNOWN. In fact, if I have any bilinear map from m times n to p. I can also define a map from e to p while it says delta l m. To F of m, n. And now, when my map F is bilinear, this map from A to P Must factor. Through the quotient, As f is bilinear. This map. Must be zero On F. So I can factor by F here, And I have such a diagram. And I also have units here because my map from e to p, is to determined by the images of m n. So this factorization is determined by images. Of delta mn. So this is unique. So I can call it five and I can identify these with. M or times n. Now what is clear about my object I just introduced? Now this is generated by those classes of delta mn. Modular m. I shall denote them m m. So with this I just delta mn, modular F, in fact any element of my product is a finite sun of such things. Well, but of course, not equal to one. This set of those combinations, sorry those [INAUDIBLE] product, so remark not equal to the set of such m [INAUDIBLE] n's because e was a free module generated by those deltas, right? Well in fact, any element from the chancellor product I can write as a finite sum Of such symbols. But I cannot reduce these further. Now, you can ask why haven't I just defined the [INAUDIBLE] product by this construction? Why am I talking of this universal property? And the answer is because it is easier to prove things this way. So advantages Of the universal property. Is as follows that the proofs become easy. For example, if I have to prove commutativity, let us prove that M tensor N Well in fact we're right over A. If it is a chance or product of two A modules, we write M chance over N over A which I have not done before but I shall do this form now on. Well, if we want to prove that this is the same as N m over a. Then it is very elegant with the universal property so indeed. The map from M times n to N times M which sends m, n to n [INAUDIBLE] m is bilinear. Therefore, it factors through the [INAUDIBLE] product. We have M tensor N to N tensor M. And to construct the n. Inverse of alpha we do just the same, well write in the same way obtain the inverse map. In the other direction So in the same way we prove, for instance, the same type of argument yields. For instance that A tensor M over A is just isomorphic to M. Well, let me prove something slightly more serious. More seriously we have seen that the chance of product is generated by those little products. So in others words, we have seen That if M is generated by e1 and so on, en, and N is generated by epsilon 1 and so on, epsilon m. Or then The tensor product is generated by those ei tensor epsilon in j. Well this is easy, which is less easy is to prove that now if m and n are free [INAUDIBLE] modules with basis you run and so on [INAUDIBLE] e1 and so one, en is a basis of m and epsilon 1 and so on, epsilon m is the basis of n. Then, ei times epsilon j, where I is between one and n, and jay is between one and m, is a business of m, change our n. On this is. Isn't it done with the universal property? Well, how we can do such a thing? So I think this deserves a name. Let's call it proposition 1. So proof. Let us define a bilinear map from M x N to A. Ascending Let's see. The sum of ai ei, sum of v g, epsilon g, to A as 0, bj0. So let me call it by a name. Let me call it, say, fi0g0. This is bilinear. So it factors through the [INAUDIBLE] product. And what happens, what happens here we have, of course, that. So, let me go left here then. Add zero to zero. This add zero to zero sends E zero chances. Epsilon j zero to one. And all the other things to zero. So if I have a linear combination. Of my transfer product. If I have the sum of alpha i j if e i tensor epsilon j which is equal to zero, then a plane This F-tilda, We see that alpha-i zero, j-zero is zero. But we can do this for any i-zero and j-zero. for all i zero, j zero. To conclude, that all coefficients are zero. So, in particular we come back to a notion, which is probably very familiar for you already, the transor product of vector spaces. So in particular, the transor product Of vector spaces, say k vector spaces with bases e 1 and so on e n and epsilon 1 and so on. Epsilon m is k vector space with base the eye Epsilon g. This is how it is often defined. One just introduces formally such a base and view sub vector spaces on base. But in general it is much better to use this universal property. [SOUND] [MUSIC]