Can anyone explain me as to why *Tensor Products* are important, and what makes Mathematician's to define them in such a manner. We already have *Direct Product*, *Semi-direct* products, so after all why do we need Tensor Product?

The Definition of **Tensor Product** at Planet Math is confusing.

**Definition:** Let $R$ be a commutative ring, and let $A,
B$ be $R$-modules. There exists an $R$-module $A\otimes B$, called the
*tensor product* of $A$ and $B$ over $R$, together with a canonical
bilinear homomorphism
$$\otimes: A\times B\rightarrow A\otimes B,$$
distinguished, up to isomorphism, by the following universal
property.
Every bilinear $R$-module homomorphism
$$\phi: A\times B\rightarrow C,$$
lifts to a unique $R$-module homomorphism
$$\tilde{\phi}: A\otimes B\rightarrow C,$$
such that
$$\phi(a,b) = \tilde{\phi}(a\otimes b)$$
for all $a\in A,\; b\in B.$

The tensor product $A\otimes B$ can be constructed by taking the free $R$-module generated by all formal symbols $$a\otimes b,\quad a\in A,\;b\in B,$$ and quotienting by the obvious bilinear relations: \begin{align*} (a_1+a_2)\otimes b &= a_1\otimes b + a_2\otimes b,\quad &&a_1,a_2\in A,\; b\in B \\ a\otimes(b_1+b_2) &= a\otimes b_1 + a\otimes b_2,\quad &&a\in A,\;b_1,b_2\in B \\ r(a\otimes b) &= (ra)\otimes b= a\otimes (rb)\quad &&a\in A,\;b\in B,\; r\in R \end{align*}

Also what do is the meaning of this statement:(**Why do we need this?**)

- Every bilinear $R$-module homomorphism $$\phi: A\times B\rightarrow C,$$ lifts to a unique $R$-module homomorphism $$\tilde{\phi}: A\otimes B\rightarrow C,$$ such that $$\phi(a,b) = \tilde{\phi}(a\otimes b)$$ for all $a\in A,\; b\in B.$

Me and my friends have planned to study this topic today. So i hope i am not wrong in asking such questions.