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Let $R$ be a commutative Noetherian ring. Let $Ab$ denote the category of abelian groups.

Let $T:R$-Mod $\to Ab$ be a covariant, additive functor such that for any sequence of $R$-modules, $A \xrightarrow{f} B \xrightarrow{g}C$, the sequence is exact if and only if $T(A) \xrightarrow{T(f)} T(B) \xrightarrow{T(g)}T(C)$ is exact.

Then, is it true that $T$ necessarily preserves either arbitrary direct sums or arbitrary direct products ? i.e. is it true that $T$ preserves either direct limit or inverse limits ? i.e. is it true that either there exists $B\in R$-Mod with $T(-)\cong -\otimes_R B$ or that there exists $P\in R$-Mod with $T(-)\cong Hom_R(P,-) $ ?

Related On faithfully flat and faithfully projective modules

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No: for instance, you can take $T$ to be a direct sum of functors of the two different types you mention. Explicitly, you could let $S$ be any infinite set and define $T(A)=A^{\oplus S}\oplus A^S$.

For a funkier example, you could take $R$ to be a field and let $T$ be the functor that takes a vector space to its double dual.

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