I have a lot of sympathies for Mergism, but in this case, it's a bit tricky. A Structurist solution might be better here. I would rather see less content and more disambiguation and sectioning on this page, with "Main article: ..." just under the heading of each section. What I don't like about this page is that it favors the set-theoretic/algebraic usage of kernel a bit too much over the other uses, such as the integral transform or statistical usages.

Although the set-theoretic usage is more general than the algebraic, it is not the case that someone trying to understand the notion in group theory or ring theory for the first time really needs the set-theoretic version. Usually, "preimage of the neutral element" will do. From this point of view, kernel_(algebra) is OK, although still a bit too chatty, particularly in the "General Case" section. (The "Mal'cev algebra" section needs help because of the redlinks.) It also underplays the category theory notion too much compared to the universal algebra notion. Again, I think that page could be improved with better sectioning. 

Null space is very problematic. It is claimed on that page that some people use both that term and the term kernel to denote the zero set of an operator, where the latter term is used in its broadest, possibly nonlinear, sense. That coincides with the algebraic version only in the case of linear operators, i.e., linear transformations. I confess I have never seen either null space or kernel used synonymously with zero set, but have only seem the terms used in the algebraic (or in the case of kernel, set-theoretic) senses. Maybe I live in a bubble. If kernel really is used sometimes in this rather weak sense, then this page needs a brief mention of it. Again, the page Null space itself is rather unstructured. Better sectioning could probably fix it.

Hmm. That makes three times I mentioned better sectioning. Maybe I really am a Structurist.