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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 the proof of Proposition 3.2.6 ...
Proposition 3.2.6 and its proof read as follows:
View attachment 8078
In the above proof of Proposition 3.2.6 we read the following:"... ... Then $$x - x'\in \text{Ker } g = \text{Im } f$$, so $$(x - f( f' (x))) - (x' - f( f' (x'))) = ( x - x') - ( f ( f'(x) ) - f ( f'(x') ) )$$
$$= ( x - x') - f ( f' ( x - x') )$$
$$\in \text{Ker } f' \cap \text{Im } f = 0$$ ... ...
Thus it follows that $$g'$$ is well-defined ... ... "Can someone please explain exactly why/how $$( x - x') - f ( f' ( x - x') ) \in \text{Ker } f' \cap \text{Im } f = 0$$ ... ... Further, can someone please explain in some detail how the above working shows that $$g'$$ is well-defined ...
Help will be much appreciated ...
Peter
Currently I am focused on Section 3.2 Exact Sequences in $$\text{Mod}_R$$ ... ...
I need some 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 8078
In the above proof of Proposition 3.2.6 we read the following:"... ... Then $$x - x'\in \text{Ker } g = \text{Im } f$$, so $$(x - f( f' (x))) - (x' - f( f' (x'))) = ( x - x') - ( f ( f'(x) ) - f ( f'(x') ) )$$
$$= ( x - x') - f ( f' ( x - x') )$$
$$\in \text{Ker } f' \cap \text{Im } f = 0$$ ... ...
Thus it follows that $$g'$$ is well-defined ... ... "Can someone please explain exactly why/how $$( x - x') - f ( f' ( x - x') ) \in \text{Ker } f' \cap \text{Im } f = 0$$ ... ... Further, can someone please explain in some detail how the above working shows that $$g'$$ is well-defined ...
Help will be much appreciated ...
Peter