Exercise 111.4.1. Let $R$ be a ring. With notation as above.

Give an example of a ring $R$ and an additive functor $F : \text{Mod}_ R \to \text{Mod}_ R$ which is not $R$-linear.

Let $N$ be an $R$-module. Show that the functor $F(M) = M \otimes _ R N$ is $R$-linear, right exact, and commutes with direct sums,

Conversely, show that any functor $F : \text{Mod}_ R \to \text{Mod}_ R$ which is $R$-linear, right exact, and commutes with direct sums is of the form $F(M) = M \otimes _ R N$ for some $R$-module $N$.

Show that if in (3) we drop the assumption that $F$ commutes with direct sums, then the conclusion no longer holds.

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