From the perspective of universal algebra, quotient structures of algebraic structures are built using congruence relations. If $A$ is an algebraic structure (a set with a bunch of operations on the set) und $R$ congruence relation on a set, then the quotient $A/R$ is well-defined and it will be an algebraic structure of the same type.
Now, as it turns out, in particular algebraic categories, these congruence relations on $A$ correspond exactly to some type of subobject of $A$. For instance, the congruence relations on a ring correspond precisely to the ideals of that ring; the congruence relations on a group correspond precisely to the normal subgroups of that group; the congruence relations on a module correspond precisely to the submodules of that module.
Why is it that the congruence relations usually correspond to some type of subobject? Is this a general phenomenon that can be generalized to all algebraic structures (as studied in this generality by universal algebra)?
Best Answer
Recall that congruences on $A$ can be viewed as certain subalgebras of its square $A^2,\,$ e.g. see here.
In algebras like groups and rings, where we can normalize $\,a = b\,$ to $\,a\!-\!b = \color{#c00}0\,$ congruences are determined by a single congruence class (e.g. an ideal in a ring). This has the effect of collapsing said relationship between congruences with subalgebras from $A^2$ down to $A.\,$ Such algebras are called ideal determined varieties and they have been much studied.
One answer to your question is that ideal-determined varieties are characterized by two properties of their congruences, namely being $\,\rm\color{#c00}{0\text{-regular}}\,$ and $\rm\color{#c00}{0\text{-permutable}}$. Below is an excerpt of one paper on related topics that yields a nice entry point into literature on this and related topics.