I'm reading Atiyah-MacDonald and trying to get a better intuition for modules. One thing that is very interesting about them so far is the variety of operations that are defined on them: $+, \oplus, \otimes, \cap, (A:B), A/B, \mathfrak{a}A, \text{Ann}(A)$, and so on. Some of these operations specifically give you back an ideal, but ideals are modules. However, the module quotient $A/B$ stands out to me as being a partial operation (in the sense that it isn't defined on every pair of modules).
The immediate motivation for this question is an exercise in Atiyah-Macdonald that asks us to prove $(N:P) = \text{Ann}((N+P)/N)$ on page 20. The exercise wants us to use the lemma on page 19 shown below.
$$ (M_1 + M_2)/M_1 \cong M_2/(M_1 \cap M_2) $$
Thinking about the RHS of this equation made me wonder why we don't just use it as the definition of $M_2/M_1$ in the first place. It's possible I'm missing something obvious.
There's a natural way to extend $A/B$ to work on any pair of modules, described below. Is there a deep reason why $A/B$ is ordinarily defined to be a partial operation?
Let $R$ be an associative, commutative, unital ring.
Let $A$ and $B$ be $R$-modules.
The module quotient $A/B$ is defined if and only if $B$ is a submodule of $A$. See Wikipedia for example.
However, it is possible to relax this restriction and allow $A$ and $B$ to be arbitrary, provided they are submodules of a common module $C$.
I'll use $A \oslash B$ to denote the "expanded" module quotient. We can define $A \oslash B$ as follows:
$$ A \oslash B = A/(B \cap A) $$
Or, equivalently, we can use the same definition that we usually use for $A/B$ and note that elements of $B$ not in $A$ do not affect the resulting congruence $\equiv$.
$$ x \equiv y \;\text{in $A \oslash B$} \iff x - y \in B $$