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Category Theory for beginners (3)

Posted: Sat Jan 16, 2016 11:18 pm
by Tsakanikas Nickos
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.
  1. 0 \( \rightarrow \) \( \displaystyle \ker (f) \) \( \rightarrow \) \( M \) \( \rightarrow \) \( \displaystyle Im(f) \) \( \rightarrow \) 0
  2. 0 \( \rightarrow \) \( \displaystyle Im(f) \) \( \rightarrow \) \( N \) \( \rightarrow \) \( \displaystyle Coker(f) \) \( \rightarrow \) 0
  3. 0 \( \rightarrow \) \( \displaystyle \ker (f) \) \( \rightarrow \) \( M \) \( \rightarrow \) \( N \) \( \rightarrow \) \( \displaystyle Coker(f) \) \( \rightarrow \) 0