My professor didn’t define what is a free abelian group $G$ (on a set $X$) but I can guess out the definition.
We can define a free abelian group $G$ on a set $X$ as a free object on $X$ in the category of abelian groups. I don’t know if my definition coincides with the mainstream definition though.
From this post, we know, strictly speaking, abelian groups and $\mathbb{Z}$-modules are not the same because Abelian groups are sets with one operation and $\mathbb{Z}$-modules are sets with two operations. However, they are isomorphic in the sense that the categories of abelian groups and $\mathbb{Z}$-modules are isomorphic.
Then two questions about the definition of free abelian groups naturally arose in my mind.
Q1: I saw people were talking about free abelian groups across this site, e.g., here, without an underlying set. I can only define a free abelian group on some set $X$. How do we define a free abelian group without an underlying set?
Q2: I also saw people were saying that free abelian groups are the same with free $\mathbb{Z}$-modules. From the point of view of set theory, they are not the same as we mentioned above. Do people actually mean that free abelian groups are isomorphic to free $\mathbb{Z}$-modules?
Thanks for help. Sorry for my long description but I just wanted to state things clear.
Q1 For every set $X$, there is an abelian group called "the free abelian group on $X$", which can be constructed explicitly as the group of formal $\mathbb{Z}$-linear combinations of elements of $X$.
If $A$ is an abelian group, we say that $A$ is "free" if there exists a set $X$ such that $A$ is isomorphic to the free abelian group on $X$.
Some abelian groups are free (e.g. the trivial group, $\mathbb{Z}$, $\mathbb{Z}^2$), and some are not (e.g. $\mathbb{Z}/2\mathbb{Z}$).
Q2 What this means is that the isomorphism $F$ between the category of abelian groups and the category of $\mathbb{Z}$-modules has the property that $A$ is free if and only if $F(A)$ is free. In other words, this isomorphism restricts to an isomorphism between the categories of free $\mathbb{Z}$-modules and free abelian groups. Or in other words, when we pretend that "abelian group" and "$\mathbb{Z}$-module" are the same thing, then the two definitions of "free" also agree.