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Currently I am focused on Section 3.2 Exact Sequences in \(\displaystyle \text{Mod}_R\) ... ...

I need some further help in order to fully understand the proof of Proposition 3.2.6 ...

Proposition 3.2.6 and its proof read as follows:

View attachment 8079

In the above proof of Proposition 3.2.6 we read the following:"... ... now define \(\displaystyle g' \ : \ M_2 \longrightarrow M\) by \(\displaystyle g'(y) = x - f(f'(x))\), where \(\displaystyle x \in M\) is such that \(\displaystyle g(x) = y\) ... ... ... ...

... ... Suppose that \(\displaystyle x' \in M\) is also such that \(\displaystyle g(x') = y\) ... ... Does the above text imply that \(\displaystyle g'(y) = x' - f( f'(x') )\) ... ... ?

Peter