Abelian Group with basis $A \times B$.

Question

I read about the following. Let $A,B$ be $\mathbb{Z}$-Modules. Then there exists an abelian group $C$ with basis $A \times B$. A R-module $D$ has a basis $\beta$, if the following mapping is an isomorphism: $$\alpha: \bigoplus_{b \in \beta} R \to D, (z_b) \mapsto \sum_{b \in \beta} z_bb.$$ I can't figure out a fitting isomorphism that has that property for the case above. I'm not sure whether I am overseeing something trivial or whether this is not too easy to see. Anyways, any help is greatly appreciated!

Answer 1

Yet, it is very simple: for any set $X$, a $\mathbf Z$-module with basis $X$ is simply the group $\mathbf Z^{(X)}$ of maps $f:X\longrightarrow \mathbf Z$ with finite support.

An obvious basis is the set of functions $\{e_x: x\in X\}$, where \begin{cases} e_x (x)=1,\\ e_x(y)=0& \text{if }y\ne x, \end{cases}
in other words the characteristic function of the singleton $\{ x \}$.

The points in $X$ and these characteristic functions are usually identified, and we speak of the ‘free abelian group with basis $X$’.