The property is given below ( from Dummit & Foote) but I have a difficulty in understanding why it is universal property and what is its importance or when usually we use it?and why the function must be bilinear to apply the universal property as mentioned in the example below? could anyone explain this for me please?
Here is an example for using it (but I do not understand why and how it is used? could anyone explain this for me please?):
Best Answer
The property expressed in Theorem 8 is an adjunction of functors; precisely the functor $S\otimes_R-\colon R\text{-Mod}\to S\text{-Mod}$ (extension of scalars) is a left adjoint to the functor $\iota\colon S\text{-Mod}\to R\text{-Mod}$ (restriction of scalars).
In a very broad sense, $R\text{-Mod}$ is “reflected” in $S\text{-Mod}$ and conversely, but something is generally lost in these correspondences. An example is $R=\mathbb{Z}$ and $S=\mathbb{Q}$; if $N=\mathbb{Z}/2\mathbb{Z}$, then $\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{Z}/2\mathbb{Z}=0$ and the theorem basically says that there is no nontrivial homomorphism of $\mathbb{Z}/2\mathbb{Z}$ into $L$, where $L$ is any $\mathbb{Q}$-module.
It is a special case of the more general adjunction between tensor functors and Hom functors.
The second part in your question has nothing to do with Theorem 8. It just states that $$ (M\oplus M')\otimes_R N\cong (M\otimes_RN)\oplus(M'\otimes_R N) $$ (and the symmetric situation when the direct sum is on the opposite side).