I often use the tensor product but I don't understand it very well despite that I know his universal property and an example of his construction, for me it is like a black box and I would like to understand it better. For example, if we consider the tensor product of 2 algebras over a field, $A \otimes_{K} B$, and if $a \in A, b \in B$, what does it imply that $a \otimes b = 0$ ?
Does someone has a good reference (a section of a book, or just lecture notes) to see more clearly what is $A \otimes_{K} B$ ?
Thank you