Group ring definition

Question

I don't understand the definition of a group ring. In "An Introduction to Group Rings" by Polcino and Sehgal (page 129), an element of the group ring $RG$ is a linear combination of elements from $G$ and coefficients from $R$ (here $R$ is a ring a $G$ is a group). In the samo book, at page 131 it says that an element of the group ring $RG$ is a mapping $\alpha:G \to R$ with $supp(\alpha)< \infty $. I don't understand the connection between these two definitions. A linear combination is a map? The value of a linear combination $\sum_{g\in G} a_g*g$ is an element from $R$?