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0¬ABC÷0 be a short exact sequence of abelian groups. Show that the following are equivalent. 1. There is a homomorphism r: B → A such that ri = id; 2. There is a homomorphism s : C → B so that ps = idc. 3. B AC, and there is an isomorphism from the given sequence to the sequence 0→A→A@C→C→0 with the obvious inclusion and projection maps. We call such exact sequences split.

0¬ABC÷0 be a short exact sequence of abelian groups. Show that the following are equivalent. 1. There is a homomorphism r: B → A such that ri = id; 2. There is a homomorphism s : C → B so that ps = idc. 3. B AC, and there is an isomorphism from the given sequence to the sequence 0→A→A@C→C→0 with the obvious inclusion and projection maps. We call such exact sequences split.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 3.4. Let 0¬ABC 0 be a short exact sequence of abelian groups. Show that the following are equivalent. 1. There is a homomorphism r: B → A such that ri = id; 2. There is a homomorphism s: C → B so that ps = idc. 3. BAC, and there is an isomorphism from the given sequence to the sequence 0→AA@C→C→0 with the obvious inclusion and projection maps. We call such exact sequences split.
Transcribed Image Text:Exercise 3.4. Let 0¬ABC 0 be a short exact sequence of abelian groups. Show that the following are equivalent. 1. There is a homomorphism r: B → A such that ri = id; 2. There is a homomorphism s: C → B so that ps = idc. 3. BAC, and there is an isomorphism from the given sequence to the sequence 0→AA@C→C→0 with the obvious inclusion and projection maps. We call such exact sequences split.
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