Is there a different construction of "the" tensor product of two modules? It may be a pseudo question. But I still decide to ask. Given two $k$-modules $M$ and $N$，it seems to me that in the literature the tensor product $M\bigotimes_kN$ is always defined as the quotient of the free module generated over the set  $M\times N$ modulo the ideal generated by the bi-linearity relations. But I am curious to see a different construction of $M\bigotimes_kN$ which is of course isomorphic to the one mentioned above.
 A: In my first encounter with the tensor product of modules (in a course on representation theory by prof. Lenstra), it was done in the following spirit:
First, the tensor product $M\otimes _RE$ is defined using the universal property. Next, we prove the following (and other) elementary properties (here I am concentrating on the object and leaving out the morphism):


*

*$M\otimes _RR$ exists, and equals $M$.

*'$\otimes$ commutes with $\oplus$': if $(M\otimes E_i)_i$ exist, then $M\otimes \oplus_iE_i$ exists and equals their direct sum.

*'$\otimes$ commutes with $coker$ (right-exactness of $M\otimes_R-$)': Let $f:E\to F$ be $R$-linear, and assume $M\otimes_RE$ and $M\otimes_RF$ exist. Then $M\otimes_R coker f$ exists and equals $coker(M\otimes_RE\xrightarrow{1\otimes f}to M\otimes_RF)$


Theorem: The tensor product $M\otimes_RE$ exists.
Proof: Take a generating set S of E, i.e. the natural map $f:R^{(S)}\to E$ is surjective. Next pick a generating set T of ker(f), so the natural map $h:R^{(T)}\to R^{(S)}$ has image ker(f). Now $coker(h)=R^{(S)}/\ker f$ is (isomorphic via f to) $E$. 
By property 1 and 2, $M\otimes_RR^{(T)}$ and $M\otimes_RR^{(S)}$ exist. By property 3, we conclude that $M\otimes_RE$ exists. 
Remark:  I guess a didactical merit of this approach (compared to the standard construction as the free abelian group on the product modulo bilinear relations) is that it forces you to think and reason in terms of the universal property and exact sequences. I am not sure if this is also the reason my teachers had in mind. 
A: A rather general definition of tensor products, far beyond the algebraic setup can be found in the arxiv papers 0808.0095, 0807.1436. A recent application to quanta is in the arxiv paper 1203.0412.
