I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with multiplicative sets. More precisely, given a commutative ring $R$, the ideal structure Ideal($R$) has a lattice structure (intersection and sums being the meet and join). Also the set of multiplicative submonoids of $R$, Mult($R$), also has a lattice structure (intersection and multiplicative closure being the meet and join). It can be verified that the following are functors, $$ K\colon R\downarrow CRing\to Ideal(R),\ (f\colon R\to S)\mapsto ker(f)\\ Q\colon Ideal(R)\to R\downarrow CRing,\ I\mapsto (\pi_I\colon R\to R/I) $$ with $\pi_I$ being the canonical projection. It can also be easily verified that $Q\dashv K$, i.e., formation of quotients is left adjoint to the formation of kernels. The counit of this adjunction is the well-known isomorphism theorem for rings. The unit is just the identity.

Similarly, if we denote the group of units of a ring $S$ by $S^*$, we can verify that the following are functorial as well, $$ L\colon R\downarrow CRing \to Mult(R),\ (f\colon R\to S)\mapsto f^{-1}(S^*)\\ M\colon Mult(R)\to R\downarrow CRing,\ S\mapsto (\ell_S\colon R\to S^{-1}R) $$ with $\ell_S$ being the canonical map $r\mapsto \frac{r}{1}$. Again, we can show $L\dashv M$, i.e., the formation of localizations is left adjoint to the formation of multiplicative sets. The counit of this adjunction is the universal mapping property of localizations. The unit being $$\eta_S\colon S\hookrightarrow ML(S)=\begin{cases}S\vee R^*& 0\not\in S\\ R& 0\in S \end{cases}.$$

(NOTE: Since $1=0$ in the zero ring, we are allowing $0$ to be a unit in the zero ring.)

My question: Is there something deeper happening here? Are the constructions above particular instances of congruences on slice categories? Or can we generalize to other algebraic structures, i.e., in a universal algebraic sense?

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    $\begingroup$ You can view $R / CRing$ as a variety of universal algebra (take the ring axioms and add in a new constant for each element of $R$ and a new axiom for each arithmetic equation about elements of $R$). I'm not sure if this connects to the other categories you're looking at. $\endgroup$ – Hurkyl Aug 23 '14 at 6:03
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    $\begingroup$ In both cases, you have a dual lattice: the lattice of quotients of $R$ and the lattice of localizations of $R$, both subcategories of $R / CRing$. It may be more fruitful to think in terms of that. $\endgroup$ – Hurkyl Aug 23 '14 at 6:13
  • $\begingroup$ I don't really understand the question. What do you want to know? $\endgroup$ – Qiaochu Yuan Aug 23 '14 at 6:16
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    $\begingroup$ You are just rephrasing the usual universal properties in terms of an adjunction, which is always possible. Nothing deep here, really. $\endgroup$ – Martin Brandenburg Sep 3 '14 at 12:51
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    $\begingroup$ @Rachmaninoff, its not overdoing it, in fact this is a very nice way of describing what's going on. $\endgroup$ – goblin Feb 11 '15 at 15:56

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