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I am reading Paul E. Bland's book "Rings and Their Modules" ...
Currently I am focused on Section 3.2 Exact Sequences in $$\text{Mod}_R$$ ... ...
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 $$f'f = \text{id}_{M_1}$$ ... ... BUT ... ... I thought that $$f'f$$ was only defined on $$f(M) = \text{Im } f $$ ... ... what then happens to elements $$x \in M$$ that are outside of $$f(M) = \text{Im } f$$ ... ... see Fig. 1 below ...
View attachment 8073Note that in the proof of Proposition 3.2.3 we read:" ... ... If $$x \in M$$, then $$f'(x) \in M_1$$ ... ... " But ... how does this work for $$x$$ outside of $$f(M) = \text{Im } f $$ such as $$x$$ shown in Fig. 1 above?
I would be grateful if someone could explain how Definition 3.2.2 "works" ... ...
Peter
Currently I am focused on Section 3.2 Exact Sequences in $$\text{Mod}_R$$ ... ...
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 $$f'f = \text{id}_{M_1}$$ ... ... BUT ... ... I thought that $$f'f$$ was only defined on $$f(M) = \text{Im } f $$ ... ... what then happens to elements $$x \in M$$ that are outside of $$f(M) = \text{Im } f$$ ... ... see Fig. 1 below ...
View attachment 8073Note that in the proof of Proposition 3.2.3 we read:" ... ... If $$x \in M$$, then $$f'(x) \in M_1$$ ... ... " But ... how does this work for $$x$$ outside of $$f(M) = \text{Im } f $$ such as $$x$$ shown in Fig. 1 above?
I would be grateful if someone could explain how Definition 3.2.2 "works" ... ...
Peter