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Can we consider the Coproduct of two objects in an abelian category as an extension of them the same as in the category of modules

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Yes, sure. If $A,B$ are objects of an abelian category (actually linear category suffices), then $0 \to A \to A \oplus B \to B \to 0$ is exact.

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Thanks alot for your answer. I would be grateful if could give me any reference for this. – Ggugl Nov 17 '13 at 13:51
Any book on abelian categories? – Martin Brandenburg Nov 17 '13 at 14:20

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