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Is there an additive functor between abelian categories, which preserves monomorphisms, but is not left exact?

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Consider an integral domain $R$ which is not a field, and choose $a \in R \setminus (\{0\} \cup R^*)$. Consider the functor $\mathsf{Mod}(R) \to \mathsf{Mod}(R)$ which maps an $R$-module $M$ to its submodule $aM$. Clearly this preserves monomorphisms, but it is not left exact: The exact sequence $0 \to R \to Q(R) \to Q(R)/R$ becomes $0 \to aR \to Q(R) \to Q(R)/R$ after applying the functor.

Alternatively, it suffices to give an additive functor between abelian categories which preserves epimorphisms, but is not right exact. Take an integral domain $R$, which is not a field, and consider the functor $\mathsf{Mod}(R) \to \mathsf{Mod}(R)$ which takes an $R$-module to its maximal torsionfree quotient, i.e. which mods out the torsion submodule. This functor preserves epimorphisms, but is not right exact: The exact sequence $R \to Q(R) \to Q(R)/R \to 0$ becomes $R \to Q(R) \to 0 \to 0$ after applying the functor.

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  • $\begingroup$ Actually, both your examples work on both sides: they preserve both monomorphisms and epimorphisms, but are neither left exact nor right exact. $\endgroup$ – Eric Wofsey Dec 3 at 4:57

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