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- Jun 22, 2012

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I need some help in understanding D&F's proof of Proposition 25, Section 10.5 (page 384) concerning split sequences.

Proposition 25 and its proof are as follows:

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**Proposition 25.**The short exact sequence \(\displaystyle 0 \longrightarrow A \stackrel{\psi}{\longrightarrow} B \stackrel{\phi}{\longrightarrow} C \longrightarrow 0 \) of R-modules is split if and only if there is an R-module homomorphism \(\displaystyle \mu \ : \ C \to B \) such that \(\displaystyle \phi \circ \mu \) is the identity map id on C. Similarly, the short exact sequence \(\displaystyle 1 \longrightarrow A \stackrel{\psi}{\longrightarrow} B \stackrel{\phi}{\longrightarrow} C \longrightarrow 1 \) of groups is split if and only if there is a group homomorphism \(\displaystyle \mu \ : \ C \to B \) such that \(\displaystyle \phi \circ \mu \) is the identity map id on C.

*Proof:*This follows directly from the definitions: if \(\displaystyle \mu \) is given then define \(\displaystyle c' = \mu (C) \subseteq B \) and if C' is given then define \(\displaystyle \mu = \phi^{-1} \ : \ C \cong C' \subseteq B \).

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Now I need some help with the proof ... to be specific ... suppose we are given \(\displaystyle \mu \ : \ C \to B \) such that \(\displaystyle \phi \circ \mu = id \) on C ... ... how, then, do we show that the short exact sequence is split ... that is, how do we show that:

\(\displaystyle B = \psi (A) \oplus C' \) ... ... ... ... (1)

for the submodule C' of B?

Following D&F we put \(\displaystyle C' = \mu (C) \subseteq B \)

... ... but how then do we establish (1)?

I would appreciate some help.

Peter