When we take the quotient of an algebraic structure $A$ with respect to a subset $S$, from what I have learned last semester, the requirements on the set $S$ doesn't seem general. For groups, the set has to be a normal subgroup; for rings with unity, the set has to be a two-sided ideal; for vector spaces, the set has to be a vector subspace. Just from their definitions, it doesn't look like there is one single property shared among them.
Inspired by the first isomorphism theorem, I see that every kernel of a homomorphism is a normal subgroup/two-sided ideal/vector subspace, so I propose the following:
For an algebraic structure $A$ and a subset $S\subseteq A$, $S$ is a normal subgroup/two-sided ideal/vector subspace of $A$ (depending on what $A$ is) iff there exist another algebraic structure $A'$ being the same type as $A$ and a homomorphism $f:A\to A'$ such that $\ker f=S$.
Would this be a good characterisation of normal subgroups/two-sided ideals/vector subspaces? And it is good to take this as a general definition for normal subgroups/two-sided ideals/vector subspaces?