Split Short Exact Sequences .... Bland, Proposition 3.2.6 .... .... Another Question ....

[Math Amateur]

I am reading Paul E. Bland's book "Rings and Their Modules" ...

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

 

[castor28]

Hi Peter,

Yes, indeed, $g'(y) = x-f(f'(x)) = x' - f(f'(x'))$, since the last equality has just been proved. This shows that $g'(y)$ in unambiguously defined; please refer to my previous post for an intuitive explanation.