This is Just a curosity.
Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($ M \otimes N$,u) (where $M \otimes N$ is a $A$-module and $u: M\times N \to M\otimes N$ is a bilinear map) with the following universal property:
Universal Property: Given any bilinear map $ \phi: M\times N \to P$ there exist a unique Linear map $f: M\otimes N \to P$ such that $\phi=f o u$.
I have seen standard construction of Tensor Product using free modules. I would like to know some more construction(or at least existence) because it's interesting to see that something can be proved in many different ways. Do you have any other nice constructions of tensor product?As I don't know much Category theory so I'm not interested in a construction using category theory but you can write the construction using category theory also because that may be helpful for other users.Thank you!
This isn't an answer, but it is too long for a comment.
The categorical perspective is actually a little interesting and subtle. Because the category of modules deals with linear maps, but tensor product are defined in terms of bilinear maps, you need a way to encode bilinearity. The standard solution is first to give the space of $R$-linear maps between two modules the structure of an $R$-module, which brings us out of the basics of category theory (where hom sets are sets) into enriched category theory, at which point we can encode the set of bilinear maps $\operatorname{Bilinear}(M\times N,Z)$ as $\hom_R(M,\hom_R(N,Z))$, at which point we phrase tensor products in terms of adjunctions (the hom-tensor adjunction), which is a very useful way for dealing formally with tensor products.
There are theorems for showing that a functor has an adjoint (like the adjoint functor theorem or special adjoint functor theorem), but I have never seen a proof that $\hom_R(M,-)$ has an adjoint that wasn't an explicit construction of the tensor product. Whether this is purely pedagogical (as people see the adjoint functor theorem long after they see tensor products), or if there are actual technical hurdles to this approach, I don't know. If I have an opportunity, I will try to work out a proof via this avenue and will edit accordingly.