homology as a functor into $R$-module notation confusion So I was reading this question
Homology is a functor into Groups or R-Modules
and now I am a bit confused.
According to the accepted answer, if we think of $\mathbb{Q}$ as an abelian group or $\mathbb{Z}$-module, we write $H_n(X;\mathbb{Q})$ and we have $H_n(X;\mathbb{Q})$ as a $\mathbb{Z}$-module.
However, we can also think of $\mathbb{Q}$ as a $\mathbb{Q}$-module, then $H_n'(X;\mathbb{Q})$ is in fact a $\mathbb{Q}$-module (this is probably not the right notation, I write $H_n'$ to differentiate it from the $\mathbb{Z}$-module $H_n$). Now since $\mathbb{Q}$ is a field, $H_n'(X;\mathbb{Q})$ is in fact a vector space.
It seems that the 2nd way of thinking we have added structure. My questions are followed: what is the right notation for $H_n'(X;\mathbb{Q})$ and is there any added benefit of thinking in the 2nd way, or they will turn out to be the same?
 A: I believe there is no notation to distinguish one from another. Let me see if I can make it clear why.
Just to make it clear the answer from the other question, it basically is saying that, generally, chain complexes are made of $R$-modules and $R$-homomorphisms for some ring $R$. In this way, each $n$th homology is a $R$-module (since it is a quotient of $R$-modules). In other words, we have a functor
$$H_n\colon Complex(R-Mod)\to R-Mod,
$$
where $Complex(R-Mod)$ denotes the category of chain complexes of $R$-modules.
So, when we think of singular homology, we just need to establish who the ring $R$ will be (i.e. from which ring we are gonna take the coefficients). And, because of that, we utilize the notation $H_n(X;R)$.
In this way, if you choose $R=\mathbb{Q}$, we will use the notation $H_n(X;\mathbb{Q})$.
As you said we can conclude that $H_n(X;\mathbb Q)$ is a $\mathbb Q$-vector space (i.e. a $\mathbb Q$-module). But we can also think of $H_n(X;\mathbb Q)$ as an abelian group, because every module (and every vector space) is an abelian group, and, therefore, it is a $\mathbb Z$-module. I don't see any reason to differentiate them in different notations as I see no reason to distinguish $\mathbb R^n$ as a $\mathbb R$-vector space from $\mathbb R^n$ as an abelian group.
For the second question, I have a similar answer: I see no benefit of thinking of $H_n(X;\mathbb Q)$ as only a $\mathbb Z$-module in the same way that I see no benefit of thinkink of $\mathbb R^n$ only as an abelian group.
