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See Correspondence theorem for groups for the statement of the theorem in case of groups.

Similar results holds when "groups" is replaced by "rings" and the bijective correspondence is between the ideals of $R/I$($I$ is an ideal of $R$) and ideals of $R$ which contain $I$. We can also find the correspondence in sets, modules and algebras.

The correspondence theorem is very universal in the sense that we can picture it very easily and it holds with varies of algebraic structures, but I have never seen any general form which unifies all the above results. Is there some insight for this theorem to help us generalize it to a more general structure, e.g. arbitrary objects in a category?

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  • $\begingroup$ I think if you want to unify the correspondence theorems for all of the algebraic structures you mentioned, you should first find a way to unify all of these algebraic structures and look at them as special cases of a more general structure which seems at least very hard to do because the nature of a non-Abelian groups is very different from a general ring or a module. However, you can unify Abelian groups, rings and $R$-modules and maybe $R$-algebras, I think. $\endgroup$ – stressed out Jul 14 '18 at 5:06
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    $\begingroup$ Well, the correspondence theorem is really about lattices (sub-group/ideal/module). What do the sub-object lattices have in common? Define a category based on this. (PS I haven't tried, but I would guess, this is really hard to do). $\endgroup$ – David Hill Jul 14 '18 at 5:55
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I think the most general thing I know about this comes from universal algebra.

In the sense of universal algebra, a type $\tau$ is a family of function symbols $f_i$ with a given (say, finite) arity $n_i, i\in I$.

An algebra $\mathbf{A}$ of type $\tau$ is a set $A$ together with functions $f_i^\mathbf{A}: A^{n_i}\to A$ for each function symbol $f_i$.

Imposing a given type and given equations yields the common "algebraic" categories you know (rings, groups, modules, lattices, etc.) .

Then one may define a congruence on an algebra $\mathbf{A}$ as an equivalence relation which respects the defining functions, e.g. for a single binary operation $m$, we ask that if $x$ is equivalent to $x'$, $y$ to $y'$, we have that $m(x,y)$ is equivalent to $m(x',y')$. I think you can guess the obvious generalization and thus definition of a congruence.

Now given a congruence $\Theta$ on $\mathbf{A}$ we may define a quotient structure $\mathbf{A}/\Theta$ in the obvious way.

There are now two clear facts : the set of congruences of $\mathbf{A}$ ordered by inclusion is a complete lattice called the congruence lattice of $\mathbf{A}$; and the congruence lattice of $\mathbf{A}/\Theta$ is isomorphic to the principal filter $[\Theta)$ of congruences of $\mathbf{A}$ that contain $\Theta$.

How does this relate to what you said ? Well it turns out tyat fo groups (where we have a binary operation corresponding to multiplication, a unary operation corresponding to inversion, and a nullary operation corresponding to the neutral element) the lattice of congruences is isomorphic to the lattice of normal subgroups (this is an easy exercise) and there is in fact an isomorphism such that quotients agree !

There's a similar result for rings and ideals, where the congruence lattice of a ring is isomorphic to the lattice of bilateral ideals (and quotients coincide !). We have the same thing with modules, etc.

All of this constitutes the basics of universal algebra.

Of course one may define congruences in an arbitray category, and quotients too, but I don't know what conditions one would want to impose to ensure that there's this correspondance.

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