Is it possible to prove that BoolAlg and CRing are regular using that Set is regular? I have been required to prove that the category of Boolean algebras and the category of commutative rings with unity are regular. The exercise also requires me to show that Set is regular, so I was under the impression that I should be able to use that to aid me in the proof that BoolAlg and CRing are regular. As a matter of fact, I managed show the finite completeness of these two categories using the existence of terminal object, equalizers and binary products (all of which can be constructed using their counterparts in Set) together with a theorem in Mac Lane's book.
However, I am struggling to show that all kernel pairs have coequalizers. And I have absolutely no clue how to show that regular epimorphisms are stable under pullbacks, other than trying to do it by hand. For example by trying to show that every epimorphism coequalizes its kernel pair. For the record, I have not had a lot of luck on that front, either because of a lack of experience or due to my inability to see the painfully obvious.
Another idea I had which would work only for BoolAlg is proving that every epimorphism is surjective as a function between sets, and then hopefully use the regularity of epimorphisms in the category of sets. I did some digging and apparently epimorphisms of Boolean algebras are indeed surjective, but I do not really know how to go about proving it. I am aware this is not going to work for CRing because there are examples of epimorphisms of rings which are not surjective.
I look forward to your suggestions and insights.
PS: now that we are at it, how exactly does one define a category of algebras in general? I know of many examples, but I do not recall ever seeing a general definition. Ironically, the term gets thrown around a lot in books and courses on category theory.
 A: 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.
