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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 [FONT=MathJax_Main]Mod[FONT=MathJax_Math]R ... ...
I need some help in order to fully understand the proof of Proposition 3.2.7 ...
Proposition 3.2.7 and its proof read as follows:
https://www.physicsforums.com/attachments/8082
In the above proof we read the following:
"... ... Then $$M_2 \cong M/ \text{ Ker } g \cong N$$ and $$ \text{ Ker } g = \text{ I am } f \cong M_1$$ ... ... My questions regarding the above are as follows:Question 1I understand that $$M_2 \cong M/ \text{ Ker } g$$ by the First Isomorphism Theorem for Modules ... ... but why is $$M_2 \cong M/ \text{ Ker } g \cong N$$ ... ... ?Question 2
Why, exactly, is $$\text{ Ker } g = \text{ I am } f \cong M_1$$ ... ... ?
Help will be much appreciated ...
Peter
Currently I am focused on Section 3.2 Exact Sequences in [FONT=MathJax_Main]Mod[FONT=MathJax_Math]R ... ...
I need some help in order to fully understand the proof of Proposition 3.2.7 ...
Proposition 3.2.7 and its proof read as follows:
https://www.physicsforums.com/attachments/8082
In the above proof we read the following:
"... ... Then $$M_2 \cong M/ \text{ Ker } g \cong N$$ and $$ \text{ Ker } g = \text{ I am } f \cong M_1$$ ... ... My questions regarding the above are as follows:Question 1I understand that $$M_2 \cong M/ \text{ Ker } g$$ by the First Isomorphism Theorem for Modules ... ... but why is $$M_2 \cong M/ \text{ Ker } g \cong N$$ ... ... ?Question 2
Why, exactly, is $$\text{ Ker } g = \text{ I am } f \cong M_1$$ ... ... ?
Help will be much appreciated ...
Peter