I have a question very similar to that asked in Free $\mathbb{Z}$-modules, but is not answered in that thread. This thread cleared up some of my confusion, but still leaves me with the question:
Why are free abelian groups equivalent to free $\mathbb{Z}$-modules rather than having a more general definition where we can specify the ring? Or am I incorrect in this interpretation?
Some more info:
In my study of homology theory, I work with tons of free modules where the scalars come from $\mathbb{Z}/p$. However, when I was first learning homology theory, the fundamental object of study was referred to as a free abelian group by every text I could find (because we were working with free $\mathbb{Z}$-modules). This has led to a lot of confusion on my part about how to refer to free $\mathbb{Z}/p$-modules, since I've want to keep calling them free abelian groups generated by a set $S$, but now with coefficients in $\mathbb{Z}/p$. I interpret the discussion in the question I link above to suggest that this is wrong and I should not use the term "free abelian group" to refer to a free $\mathbb{Z}/p$-module. Is this the case? If so, is there an analogous way to refer to a free $\mathbb{Z}/p$-module that emphasizes "abelian group" in the same way that we can refer to a free $\mathbb{Z}$-module as a free abelian group? Or should I just stick with using "free $\mathbb{Z}/p$-module"?
Best Answer
A free abelian group is a free $\mathbb{Z}$-module. That's because an abelian group is essentially the same thing as a $\mathbb{Z}$-module: every abelian group can be made into a $\mathbb{Z}$-module in exactly one way (and homomorphisms of abelian groups are the same as homomorphisms of $\mathbb{Z}$-modules). A module over some other ring $R$ always has an underlying abelian group structure, but if $R$ is different from $\mathbb{Z}$, then not every abelian group has a unique $R$-module structure, so $R$-modules are not the same as abelian groups in the same way $\mathbb{Z}$-modules are.
If you have a module $M$ over some ring $R$, then calling $M$ a free abelian group would mean that the underlying abelian group of $M$ (when you forget about the scalar multiplication of $R$) is a free $\mathbb{Z}$-module, not that $M$ is a free $R$-module. So, for instance, a free $\mathbb{Z}/p$-module should definitely not be called a free abelian group.
There isn't any special name like "free abelian group" for free modules over any ring other that $\mathbb{Z}$. If you want to talk about free $\mathbb{Z}/p$-modules, you should just call them free $\mathbb{Z}/p$-modules. (Or you could just call them $\mathbb{Z}/p$-modules, since every $\mathbb{Z}/p$-module is free, but it makes sense to say "free" if you are talking about a module which is free over some specific set of generators.)