Suppose $T$ is a covariant functor from the category $R$-mod to Ab (the category of abelian groups)
Is is necessary that $T(B \oplus C) \cong T(B) \oplus T(C) $ ?
Does the answer change if we assume that the functor is left exact ?
I am trying to prove that left exact covariant preserves pullbacks and need this there. Thanks
In an additive category (in particular abelian), direct sums can be characterized by the existence of certain morphisms. So, if $A$ and $B$ are objects in the additive category and $C$ is another object, then $C$ is the coproduct and the product of $A$ and $B$ if and only if there are morphisms \begin{align} i_A\colon A&\to C\\ i_B\colon B&\to C\\ p_A\colon C&\to A\\ p_B\colon C&\to B \end{align} such that $$ p_Ai_A=1_A,\quad p_Bi_B=1_B,\quad i_Ap_A+i_Bp_B=1_C $$ Therefore, any additive functor preserves finite coproducts (and products, since they coincide).
Also, a functor between additive categories is additive if and only if it preserves finite coproducts. See Mac Lane's “Categories for the working mathematician”.