Is it possible to prove that BoolAlg and CRing are regular using that Set is regular?

Solution 1:

You're on the right track. What's happening here is that the categories of commutative rings and boolean algebras are "concrete", e.g. have sets with certain additional structure for its objects, and morphisms of sets preserving this structure as morphisms. In such a situation, being able to give the appropriate structure to constructions in Set often (but not always) gives the appropriate construction in the concrete category.

To construct the coequalizers of kernel pairs, note that the kernel pair of $X\xrightarrow fY$ is the set $K[f]=\{(x_1,x_2)\in X\times X:f(x_1)=f(x_2)\}$, which is an equivalence relation with appropriate structure (ring or Boolean algebra), and its coequalizer as a set is the quotient by this equivalence relation. If you can determine appropriate structure on the set of equivalence classes (e.g. as a ring or a Boolean algebra), and if you define it so that operations on equivalence classes depend only on the representatives, then you'll be pretty much done. (If I remember right, the theory of "category of algebras" shows this always works).

Finally, pullback-stability of regular epimorphisms will follow from the case of Set because what you will have shown is that being a regular epimorphism in rings or Boolean algebras is the same as being a regular epimorphism in Set.