It has been stated as a definition in my book that if $A$ and $B$ are $R$-algebras, the tensor product module $A \otimes B$ has a well-defined multiplication defined by
$$(a \otimes b)(a' \otimes b')=aa'\otimes bb',\ a,a' \in A,\ b, b' \in B.$$
This makes $A \otimes B$ into an $R$-algebra, called the tensor product of the algebras $A$ and $B$.
How is that map well-defined? I know somehow I have to use the universal property of tensor product. In order to do that I need a bilinear map which induces a linear map and that linear map can be extended to the desired map. But what is that bilinear map that will serve my purpose? I have tried constructing it by first fixing two elements let's say $a \in A$ and $b \in B$. Then I define a map from $A \times B$ to $A \otimes B$ by sending $(x,y)$ to $ax \otimes by$. Clearly this map is bilinear. That will induce a $R$-linear map from $A \otimes B$ to $A \otimes B$ defined by $x \otimes y \mapsto ax \otimes by$. Does that help? If so how?
Please help me.
Thank you very much.