Still trying to understand tensor. In the Hungerford Grad. Algebra book (where sigma notation is the direct sum with finite nonzero components), tensor product of modules is defined as follows.
Let $A$ and $B$ be, respectively, right and left $R$-modules. Let $F$ be the free abelian group on $A \times B$. Let $K$ be the subgroup of $F$ generated by elements of form: (with $a,a'\in A,b,b'\in B,r\in R$)
- $(a+a',b)-(a,b)-(a',b)$;
- $(a,b+b')-(a,b)-(a,b')$;
- $(ar,b)-(a,rb)$
And define $A\otimes B:=F/K$.
To prove the distributivity $$(\sum_{i\in I} A_i)\otimes B\simeq \sum_{i\in I}(A_i\otimes B),$$
the proof suggests the induced homomorphism $\alpha: \sum_{i\in I}(A_i\otimes B)\rightarrow (\sum_{i\in I} A_i)\otimes B$, which is based on Theorem I.8.5 of the book, that eventually satisfies $\alpha(\{a_i\otimes b\})=(\sum_{i\in I_0} \iota_i(a_i))\otimes b$, where $\iota_i$ is the canonical injection, and $I_0$ is the set of indices with nonzero components.
My question: Aren't the elements in $A_i\otimes B$ not always of form $a_i \times b$? And aren't those elements expressed, but not uniquely, as linear combinations of that form? Because from what I read, the induced hom is about mapping the direct product element $\{x_i\}$ to the sum of images under the canonical injection of the nonzero components, but in this case the component is not simply $a_i \otimes b$, is it?
I know $\alpha$ exists, but I am just confused about its formula as it is needed in the next step.
Thanks in advance.