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I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.2 Free Modules ... ...
I need help with some aspects of the proof of Proposition 2.2.6 ...
Proposition 2.2.6 and its proof read as follows:
View attachment 8221
View attachment 8222
Near the end of Bland's proof we read the following:
" ... ... If $$x \in M$$, then $$x$$ can be written as $$x = \sum_{ \Delta } x_\alpha a_\alpha$$, where $$a_\alpha = 0$$ for almost all $$\alpha \in \Delta$$. It follows that $$( a_\alpha ) \in R_{ ( \Delta ) }$$, so $$f (( a_\alpha )) = \sum_{ \Delta } x_\alpha a_\alpha$$ and $$f$$ is an epimorphism. ... ... "My questions are as follows:Question 1
Why/how exactly does it follow from $$x \in M$$ and $$x = \sum_{ \Delta } x_\alpha a_\alpha$$ that $$( a_\alpha ) \in R_{ ( \Delta ) }$$ ... ?
Question 2
In the above quote Bland writes $$f (( a_\alpha )) = \sum_{ \Delta } x_\alpha a_\alpha$$ ... but what is going on ... ? At the start of the proof he defined $$f$$ this way ... but if he is not relying on this definition how does he calculate/formulate $$f$$ ...Question 3
Why/how exactly is $$f$$ an epimorphism ...Help will be appreciated ...
Peter
I need help with some aspects of the proof of Proposition 2.2.6 ...
Proposition 2.2.6 and its proof read as follows:
View attachment 8221
View attachment 8222
Near the end of Bland's proof we read the following:
" ... ... If $$x \in M$$, then $$x$$ can be written as $$x = \sum_{ \Delta } x_\alpha a_\alpha$$, where $$a_\alpha = 0$$ for almost all $$\alpha \in \Delta$$. It follows that $$( a_\alpha ) \in R_{ ( \Delta ) }$$, so $$f (( a_\alpha )) = \sum_{ \Delta } x_\alpha a_\alpha$$ and $$f$$ is an epimorphism. ... ... "My questions are as follows:Question 1
Why/how exactly does it follow from $$x \in M$$ and $$x = \sum_{ \Delta } x_\alpha a_\alpha$$ that $$( a_\alpha ) \in R_{ ( \Delta ) }$$ ... ?
Question 2
In the above quote Bland writes $$f (( a_\alpha )) = \sum_{ \Delta } x_\alpha a_\alpha$$ ... but what is going on ... ? At the start of the proof he defined $$f$$ this way ... but if he is not relying on this definition how does he calculate/formulate $$f$$ ...Question 3
Why/how exactly is $$f$$ an epimorphism ...Help will be appreciated ...
Peter