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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?

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    Yes, kernels are normal subgroups etc. But I think, such a characterization would not help too much as a definition, because, say, the usual normal subgroup definition is much more "explicit" and very nice as it is. – 2017-01-20
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    The notions of the "right objects" to quotient out by differs from category to category but the ideas are always the same -- you want the quotient object to be an actual object of the category! Explicitly, $G/N$ is a group under the canonical operation inherited from $G$ if and only if $N$ is normal! Normal subgroups are the definition that do the trick here. Same for every other example. – 2017-01-20
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    I really like your definition! – 2017-12-29

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