I have to show that $(\mathbb{Z}[G],+,.)$ is a unitary ring, where $$\mathbb Z[G]=\{\sum_{g \in G} a_g.g| a_g \in \mathbb Z, a_g \neq 0, \text{only for finite g in G}\}$$ with $G$ group and $(\sum a_g.g)+(\sum b_g.g)=\sum (a_g+b_g).g$;$(\sum a_g.g).(\sum b_g.g)=\sum(\sum_{g_1 g_2=g}a_{g_1}b_{g_2}).g$.
I am confused about how $\mathbb Z[G]$ is defined. If $a_g \in \mathbb Z$ and $g \in G$, does it make any sense to do the operation $a_g.g$? I mean, is $a_g.g$ in $\mathbb Z$ or in $G$?. I would appreciate if someone could describe me this ring and to tell how it is called in case it has a particular name. Thanks in advance.
$\Bbb Z[G]$, or sometimes just $\Bbb ZG$, is called the group ring (of $G$ over $\Bbb Z$). One can think of the ring as the free $\Bbb Z$-module generated by $G$: $$\Bbb Z[G]=\bigoplus_{g\in G}\Bbb Zg.$$ Then one can equipe $\Bbb Z[G]$ with the multiplication as mentioned in the question.
Recall that one can think of elements of $\Bbb Z[G]$ as a collection $\{a_g:g\in G\}$, where $a_g$ are integers and only finitely many of $n_g$'s are non-zero. This element corresponds to $\sum a_g\cdot g$ in the question. In particular, $a_g\cdot g$ is an element of $\Bbb Zg$.
Let $e$ be the unit of $G$, then the unit of $\Bbb Z[G]$ is the element $1\cdot e\in \Bbb Z e$.