If we have a commutative ring $R$ with identity, then all of the functions from $R$ to $R$ is defined as: $$f(x)=\sum_{p \epsilon R}I_{\{p\}}(x)f(p)$$ (I have a related post here validating this series: Expressing real function algebraically for every point in its domain)
If the indicator function $I_{\{p\}}(x)$ is expressible by the operations in the ring $R$, then that should mean that all functions in $R$ are expressible in the operations in $R$ (A simple statement that follows from the series definition above).
So my question is: What branch of math most closely contains this idea? Is there a generalization?
To me, it seems that the concept of "expressibility" is related to some logic branch of math. However, i present the above series in the context of algebra.
Based on what we are talking about in the comments, there needs to be some mechanism to make the sums well-defined, and one way I suggested was to restrict to functions of finite support. So I think the best match is:
Group rings
For a group $G$ and a ring $R$, you can consider the set of functions $G\to R$ of finite support, and collectively they form a ring under certain operations.
The group elements are indeed "indicator functions" for elements of $G$.