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

I need some help in order to fully understand Definition 3.2.2 and Proposition 3.2.3 ...

Definition 3.2.2 and Proposition 3.2.3 read as follows:

https://www.physicsforums.com/attachments/8072

In Definition 3.2.2 we read that \(\displaystyle f'f = \text{id}_{M_1}\) ... ... BUT ... ... I thought that \(\displaystyle f'f\) was only defined on \(\displaystyle f(M) = \text{Im } f \) ... ... what then happens to elements \(\displaystyle x \in M\) that are outside of \(\displaystyle f(M) = \text{Im } f\) ... ... see Fig. 1 below ...

View attachment 8073Note that in the proof of Proposition 3.2.3 we read:" ... ... If \(\displaystyle x \in M\), then \(\displaystyle f'(x) \in M_1\) ... ... " But ... how does this work for \(\displaystyle x\) outside of \(\displaystyle f(M) = \text{Im } f \) such as \(\displaystyle x\) shown in Fig. 1 above?

I would be grateful if someone could explain how Definition 3.2.2 "works" ... ...

Peter