I am currently reading Künneth tensor formula over arbitrary ring, and try to find out the structure of connecting homomorphism appears in the homology long exact sequence. At some point, I have stucked, and I can completely describe the connecting homomorphism if, once, I show the following. I don't claim the following is true, but, I guess it should be true.
Let $A,B$ be two right $R$-modules, and $f:A\to B$ be $R$-linear. Also, let $C$ be a flat left $R$-module and $Z$ be a $R$-flat submodule of $C$. Write $i:Z\hookrightarrow C$ for the inclusion map. Now, consider the map $f\otimes \text{Id}_Z:A\otimes Z\to B\otimes Z$. Suppose, $c\in C$ and $a\in A$ are given, so that $f(a)\otimes c\in B\otimes Z$.
My question, is it possible to write the element $f(a)\otimes c$ as $$f(a)\otimes c=\sum_\text{finite} f(a_i)\otimes z_i,\text{ where }a_i\in A\text{ and }z_i\in Z.$$
Any help will be appreciated. Thanks in advance.