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

-module over

. But

is supposed to be an abelian group in its own right, and I don't understand how.

ANY module, whether free or not, is an abelian group.

The example in page 339 already dealt with this important case
For example, one idea would be to say that since

and

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

?

Coordinatewise, of course. This is dealt with in page 353.
Here's what my textbook has to say on the subject:

link