Consider the sequence {0} ZZ6 Z3 {0} Z-modules Z2,Z3,Z6 and Z-module homomorphisms f and...

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Consider the sequence {0} → Z₂Z6 Z3 → {0} Z-modules Z2,Z3,Z6 and Z-module homomorphisms f and g, where f(m) = 3m for m € Z₂ and g(n) 2n mod 3 for n E Ze. Show that the sequence is a split exact sequence. = [20 marks] Let R be a ring. Let {0} → MMM" → {0} be an exact sequence of R-modules. Suppose M is noetherian. Show that M' and M" are also noetherian. [20 marks] Consider the sequence {0} → Z₂Z6 Z3 → {0} Z-modules Z2,Z3,Z6 and Z-module homomorphisms f and g, where f(m) = 3m for m € Z₂ and g(n) 2n mod 3 for n E Ze. Show that the sequence is a split exact sequence. = [20 marks] Let R be a ring. Let {0} → MMM" → {0} be an exact sequence of R-modules. Suppose M is noetherian. Show that M' and M" are also noetherian. [20 marks]

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To show that the sequence 0 Z 0 is a split exact sequence we need to demonstrate the following 1 The maps f Z 0 and g 0 Z are homomorphisms 2 The comp... View the full answer