TITLE: Generalization of results from specific algebraic theories to Universal Algebra
QUESTION [7 upvotes]: I'm relatively new to universal algebra, but it seems that lots of theorems from specific algebraic theories (groups, rings) can be stated in the context of universal algebra, perhaps I'm wrong.
Examples being Jordan-Holder, Lasker-Noether, Hilbert Nullstellensatz (maybe not?) and many more, where one can talk about sobobjects, and congruences instead of "subgroups" and "ideals".
Sometimes (obviously) those theorems fail in the universal context, but some of the results stand, and even more so: We might be able to measure the failure of those theorems in the universal context.
I hardly see those kind of deep results in introductions to universal algebra even though they can give a strong "justification" for the study of the field.
Therefore I have a few questions:

Is there a specific reason not to generalize those results?
Is there a reason such results are not thought in Universal Algebra introductions (at least all the ones I found)?
Is these sort of generalizations have open research problems that cause difficulties (relates to the first question)?

REPLY [3 votes]: Many of the results you mention form part of the categorical  approach to (parts of) Universal Algebra. There is a lot of active research in this area, especially in the study of semi-abelian categories, Malt'sev categories, etc. Although this is not main stream Universal algebra and some universal algebraists may not feel it deserves mention as part of that subject, it may help to look at the book by Bourn and Borceux mentioned in the n-Lab article on semi-abelian categories.