An example of tensor product Let 
$$
\otimes:R\times R\rightarrow W
$$
$$
f:R\times R\rightarrow R~~,~f(X,Y)=XY
$$ 
$\otimes$ is tensor product, $W$ is a vector space, and $f$ is a bilinear map. As I know , we need to find a linear map $g:W\rightarrow R$ such that $f=g\circ \otimes$ . Because I can define 
$$
g(X\otimes Y) = f(X,Y)   ~~,~~ X,Y\in R
$$
then $W$ is 
$$
W=\{XY | X,Y\in R \} =R
$$
Is this is a right example about tensors?
I am fuzzy with tensor products. So I try to make a specific example. But I am not sure whether there are mistakes.
 A: Your question clearly is:

$$ g(X\otimes Y) = f(X,Y)   ~~,~~ X,Y\in R $$
  then $W$ is  $$ W=\{XY | X,Y\in R \} =R $$ Is this is a right example
  about tensors?

Answer:
(I'll assume $R = \mathbb R$)
This is a wrong statement of a valid example about tensors, but can be fixed by considering formal subtleties. $W$ is not equal to $R$, it is actually isomorphic to $R$. The basic reason is that you chose $W$ to be the codomain of a generic tensor product you denote with $\otimes$. But of course you can specify you're taking the $\otimes$ map to be the ordinary multiplication map in $R$ directly, but that's not strictly the abstract tensor product formulation.
Let us clarify this.
What you denote by $W$ is the space of linear combinations of the image of a generic universal bilinear map $\otimes$. $W$ is usually denoted $R\otimes R$. You can think of it as:
$$W = R\otimes R = \left\{\sum X\otimes Y,~~~X,Y\in R \right\}.$$
We usually omit these sums since most things of interest are linear on $W$ or defined as to be linear, so we can focus on defining nontrivial rules at each component of the sum (simple or pure tensor products).
The full map $g$ is constructed as a linear extension of the rule 
$$g(X\otimes Y) = f(X,Y) = XY$$
which takes 
$$t = \sum X\otimes Y \in R\otimes R$$ 
to $$g(t) = g\left(\sum X\otimes Y\right) = \sum g(X\otimes Y) = \sum f(X,Y) = \sum XY.$$
But $t = \sum X\otimes Y = \sum XY (1\otimes 1) = (\sum XY) 1\otimes 1$ since $\otimes$ is bilinear. Hence, to map to your notation as much as possible, 
$$
W = \left\{\sum X\otimes Y\right\} = \left\{\left(\sum XY\right) (1\otimes 1)\,|\,X,Y\in R\right\}, 
$$
which is isomorphic to the set you mentioned which is $R$.
A: I think this is what you're after:
If $R$ is a ring, then the map $f : R \times R \to R$ given by $f(x, y) = x y$ is $R$-bilinear, so it induces a linear map $\tilde{f}:R\otimes_R R \to R$ given by $\tilde{f}(x \otimes y) = f(x, y) = x y$.
In general, tensor products of modules (or, more concretely, vector spaces) turn bilinear maps $f : A \times B \to C$ into linear maps $\tilde{f} : A\otimes B \to C$ via the formula $\tilde{f}(x\otimes y) = f(x, y)$.
