Recently I am learning about homology theory from the point of view of $R$-modules. However, I can not find a satisfactory proof for it. I have found a proof using the Barratt-Whitehead Lemma but could not find the proof of this lemma. Also tried to generalize the zig-zag Lemma for homology groups to $R$-modules doing diagram chasing but I couldn't find the way to proof that the induced homomorphism $\partial_*$ that we defined for groups is a homomorphism of $R$-modules. Can someone suggest me any way to prove it or a useful resource? Thanks!
EDIT: As I couldn't find any satisfactory proof for the Mayer-Vietoris sequence, here is the process I followed from Munkres' Algebraic Topology to proof zig-zag lemma to try to do it by myself. This is the exact sequence I have: $$ 0 \to A \overset{f}{\to} B \overset{g}{\to} C \to 0 $$ where $A = \{A_n,\partial_A\}, B = \{B_n,\partial_B\}$ and $C = \{C_n,\partial_C\}$ are complex chains of $R$-modules. So to prove the zig-zag lemma, first I define $\partial_*([c_p]) = [a_{p-1}]$ where $c_p \in C_p$, $a_{p-1} \in A_{p-1}$ and $[\ ]$ denote the homology class. Using the diagram chasing argument, I had no problem to prove that $\partial_*$ is well defined.
My problem comes when trying to figure out if it is an $R$-module homomorphism. To do it, Munkres asserts that as $g(b_p + b_p') = c_p + c_p'$ and $f(a_{p-1}+ a_{p-1}') = \partial_B(b_p+ b_p')$ hold, we can say that $\partial_* [c_p + c_p'] = [a_{p-1} + a_{p-1}']$ by definition and as a consequence, $\partial_* [c_p + c_p'] = \partial_*[c_{p}] + \partial_*[c_{p}']$. I find this part of the proof a bit messy and cannot fully understand it. Finally, I cannot see how to prove that $\partial_* [\lambda c_p] = \lambda \partial_*[c_{p}]$ for any $\lambda \in R$. I guess the argument is almost the same but I get lost and cannot see it clearly.