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I am reading Paul E. Bland's book, "Rings and Their Modules".
I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need yet further help in order to fully understand the proof of Proposition 4.3.14 ... ...
Proposition 4.3.14 reads as follows:
View attachment 8323
View attachment 8324
In the above proof by Bland we read the following:
" ... ... If $$a_n \neq 0$$, let $$y = x_1 a_1 + x_2 a_2 + \ ... \ ... \ + x_{n-1} a_{n-1}$$, so that $$x = y + x_n a_n$$. If $$y = 0$$, then $$x = x_n a_n$$, so $$a_n$$ is a unit since $$x$$ is primitive. Thus $$\{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}$$ is a basis for $$F$$. ... ..."Can someone please explain exactly why/how $$\{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}$$ is a basis for $$F$$ ... ...
Help will be much appreciated ... ...
Peter==========================================================================================
It may help MHB
members reading this post to have access to Bland's definition of 'primitive element of a module' ... especially as it seems to me that the definition is a bit unusual ... so I am providing the same as follows:
https://www.physicsforums.com/attachments/8325
Hope that helps ...
Peter
I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need yet further help in order to fully understand the proof of Proposition 4.3.14 ... ...
Proposition 4.3.14 reads as follows:
View attachment 8323
View attachment 8324
In the above proof by Bland we read the following:
" ... ... If $$a_n \neq 0$$, let $$y = x_1 a_1 + x_2 a_2 + \ ... \ ... \ + x_{n-1} a_{n-1}$$, so that $$x = y + x_n a_n$$. If $$y = 0$$, then $$x = x_n a_n$$, so $$a_n$$ is a unit since $$x$$ is primitive. Thus $$\{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}$$ is a basis for $$F$$. ... ..."Can someone please explain exactly why/how $$\{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}$$ is a basis for $$F$$ ... ...
Help will be much appreciated ... ...
Peter==========================================================================================
It may help MHB
members reading this post to have access to Bland's definition of 'primitive element of a module' ... especially as it seems to me that the definition is a bit unusual ... so I am providing the same as follows:
https://www.physicsforums.com/attachments/8325
Hope that helps ...
Peter
Last edited: