Unfortunately I can't check that
belongs to the denominator, so to speak, of the underlying quotient group, because I don't know how to perform operations on that quotient group. In particular, I don't know how to perform operations on the numerator of that quotient group, which is perhaps the SOURCE of my problem in understanding the definition of a tensor product.
According to my definition, we construct
by considering the free
is supposed to be an abelian group in its own right, and I don't understand how. For example, one idea would be to say that since
are both abelian groups individually, then we can just carry over the addition operations from each of them so that
. Yet clearly this is not the case. But if not that, then how do we perform operations inside the group
Here's what my textbook has to say on the subject: link