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Let $B_1 \stackrel{f}{\to} B \stackrel{g}{\to} B_2 \to 0$ be an exact sequence. For any module $M$ the sequence $$0 \to \mbox{Hom}(B_2,M) \stackrel{g*}{\to} \mbox{Hom}(B,M) \stackrel{f*}{\to} \mbox{Hom}(B_1,M)$$ is exact.

Could anyone tell me if its true that $$ \mbox{Im }g^*=\{a \in \mbox{Hom}(B,M):\ker g \subseteq \ker a\}$$

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Yes, this is the fundamental theorem on homomorphisms.

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