Let \( \displaystyle f : M \longrightarrow N \) be an \( R \)-module homomorphism. Identify each \(R\)-module homomorphism in each of the following sequences and then show that each sequence is exact.

- 0 \( \rightarrow \) \( \displaystyle \ker (f) \) \( \rightarrow \) \( M \) \( \rightarrow \) \( \displaystyle Im(f) \) \( \rightarrow \) 0

- 0 \( \rightarrow \) \( \displaystyle Im(f) \) \( \rightarrow \) \( N \) \( \rightarrow \) \( \displaystyle Coker(f) \) \( \rightarrow \) 0

- 0 \( \rightarrow \) \( \displaystyle \ker (f) \) \( \rightarrow \) \( M \) \( \rightarrow \) \( N \) \( \rightarrow \) \( \displaystyle Coker(f) \) \( \rightarrow \) 0