Let $0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0$ be a split-exact short exact sequence of $R$-modules, where $R$ is any ring. Let $T$ be an additive functor from $R$-modules to abelian groups. Then is it true that we still get a split-exact short exact sequence $0\rightarrow TA'\rightarrow TA\rightarrow TA''\rightarrow 0$ of abelian groups? (I just don't understand the reason why the zeros at both ends are preserved by $T$)
Split exact sequences
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homological-algebra
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2A split short exact sequence is preserved by any additive functor between any two abelian categories whatsoever, because being a split short exact sequence is an equational condition. – 2012-02-29