This actually causes a lot of confusion. For a finite group $G$ and a commutative unit ring $R$ I’m trying to prove that $h\in R[G]$ defined as $h=\sum\limits_{g\in G}g$ is in the center of $R[G]$. How do you write $h$ as an element of $Maps(G,R)$? How do you interpret the coefficients $c_g$ in $R[G]=\{\sum\limits_{g\in G} c_g\cdot g \ :\ c_g\in R\}$ relative to an $h\in R[G]$?
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How do you interpret the coefficients $c_g$ in $R[G]=\{\sum\limits_{g\in G} c_g\cdot g \ :\ c_g\in R\}$ relative to an $h\in R[G]$?
The coefficients are part of the definition of a group ring. A group ring $R[G]$ is the set of linear combinations of elements of the group, with coefficients in the ring, endowed with the natural operations (notably the product in $R[G]$ combines the product in $R$ and that in $G$).
$$h=\sum\limits_{g\in G}g$$ How do you write $h$ as an element of $Maps(G,R)$?
Each element of $R[G]$ is characterised by the coefficient in front of each element of $G$. Therefore $\sum_\limits{g\in G}c_g\cdot g$ can be thought of as the map $g\mapsto c_g$.
The notation $f_g$ is just an alternative functional notation for $f(g)$. That's what the comment is referring to.
The element $h=\sum_{g\in G} g=\sum _{g\in G}1\cdot g$ can be interpreted as a map $G\to R$ by saying that $g\mapsto 1$ for all $g\in G$.
The coefficients are simply a listing of where the map sends each $g\in G$. So if you know the coefficients, you know the map, and if you know the map you know the coefficients.
Is it absolutely necessary to prove it's in the center via the map interpretation? Isn't it much easier to notice that multiplying $gh=h=hg$ for all $g\in G$? Then since $h$ commutes with everything in $R$ and $G$, it commutes with everything in $R[G]$.
(Personally I've never got any use out of the functional interpretation, but then again I don't have a lot of representation theory training.)