Let $S_1,S_2\subseteq \mathbb{R}$. Given two functions $f_1\colon S_1\to \mathbb{R}$ and $f_2\colon S_2\to \mathbb{R}$, we can define a new function $f_1+f_2\colon S_1\cap S_2\to \mathbb{R}$ by the rule $(f_1+f_2)(x)=f_1(x)+f_2(x)$. In this way, we have a binary operation on the set $PF(\mathbb{R})$ of partial $\mathbb{R}$-functions. We can similarly define multiplication. The resulting structure, $PF(\mathbb{R},+,\cdot)$, is almost a ring, except that we don't have additive inverses. (Note that $-\sqrt{x}+\sqrt{x}$ isn't the additive identity, because it is only defined when $x\geq 0$.)
On the other hand, if we limit ourselves to a given domain $S\subseteq \mathbb{R}$, and we then look at the set $F(S)=\{f\colon S\to \mathbb{R}\}$, then this set is a ring under these operations. The downside here is that we force all functions to have domain $S$, even if they are perfectly defined on a bigger domain. (On the upside, we don't have to differentiate between restricted versions of the same function.)
I'm requesting references to any research on the dichotomy between these two viewpoints. In particular, I'm interested to know more information about the algebra structure $PF(\mathbb{R},+,\cdot)$, and for opinions on the pedagogy between these two viewpoints.
