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Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q \cap N = 0$.

I've got just a short question what maximality means here: Does it mean that every other $Q'$ with $Q' \cap N = 0$ is contained in $Q$, or does it mean that every $Q'$ with $Q' \cap N = 0$ and $Q \subseteq Q'$ satisfies $Q = Q'$? Usually the latter is meant by maximal, right? However, then I don't understand at all the proof (sketches) in the section on essential monomorphisms and injective hulls in Gabriel's thesis.

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Apart from 'category-theory', the tags are quite unrelated to the question. Why not simply 'abelian-categories'? – Mariano Suárez-Alvarez Sep 1 '10 at 14:35
Here I haven't got enough rep. yet to add new tags. – Martin Brandenburg Sep 1 '10 at 15:34
up vote 1 down vote accepted

"Maximal" means the latter, i.e. if $Q'$ contains $Q$ and $Q'\cap N = 0$, then $Q' = Q$. Since complements are very seldom unique (as one knows already from the case of subspaces of vector spaces), the former condition is prohibitively strong, and would almost never be realized in practice.

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