Expressing real function algebraically for every point in its domain

Question

Let's say that f(x) is a function with its domain called $A$ with its codomain labeled $B$. The indicator function $I_{\{p\}}(x)$ has a value of the multiplicative identity, $1$, when $x=p$, and $0$ otherwise. With these definitions, does the expression below hold true in a commutative ring? $$f(x)=\sum_{p\in A} I_{\{p\}}(x)f(p)$$ If this is true, then this can be considered a generalization of the commutative boolean ring theorem called Boole's expansion theorem (assuming that such a ring exist with addition and multiplication operations in a commutative ring) $$g(a)=\overline{a}g(0) + ag(1)$$

Answer 1

The equation is true if we agree that a sum with only finitely many non-zero summands is the um of those finitely many summands. (And of course, $B$ must at least be an abelian group )