Axioms of a Ring and a Simple Example (that confused me) The professor put an example of a ring on the board today:

Let $G$ be a finite group and define $RG = (a_1g_1+a_2g_2 + ... a_ig_i |a_i \in \mathbb{Z})$.

She then gave an example of multiplaction of "monomials" in this ring:
$2g_i5g_j=10g_k$ where $g_ig_j=g_k$. This example was confusing to me becuase a ring is one set with 2 operations. But here we're using 2 sets ($\mathbb{Z}$ and $G$). So What rules of the ring are allowing me to commute $5$ and $g_i$? 

You should be thinking of an element $a_i g_i$ (where $a_i \in \mathbb{Z}$) as the sum of $g_i$ with itself $a_i$ times. So this element is really
$$
2g_i 5g_j = (g_i + g_i)(g_j + g_j + g_j + g_j + g_j)
$$
and using the basic ring axioms, you can see that this expands to $g_{i}g_j$ added to itself $10$ times.
In a more "categorical" sense, and perhaps this won't make sense to you now, but what this says is that for every ring $R$ there is a unique ring morphism $\mathbb{Z} \to R$ which lands in the center of $R$. Namely, I must send $1 \in \mathbb{Z}$ to $1 \in R$, and this uniquely determines the ring morphism. Since this lands in the center of $R$, this tells you why the $5$ and the $g_i$ commute.
