 #1
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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 some help in order to fully understand the proof of Proposition 4.3.14 ... ...
Proposition 4.3.14 reads as follows:
In the above proof by Bland we read the following:
" ... ... The induction hypothesis gives a basis ## \{ x, x'_2, \ ... \ ... \ x'_{n 1} \}## of ##M## and it follows that ## \{ x, x'_2, \ ... \ ... \ x'_{n  1}, x'_n \}## is a basis of ##F## that contains ##x## ... ... "
My question is as follows:
Why/how exactly does it follow that ##\{ x, x'_2, \ ... \ ... \ x'_{n  1}, x'_n \}## is a basis of ##F## that contains ##x##. ... ...
Help will be appreciated ...
Peter
====================================================================================================
It may help PFmembers 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:
Hope that helps ... ...
Peter
I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.14 ... ...
Proposition 4.3.14 reads as follows:
In the above proof by Bland we read the following:
" ... ... The induction hypothesis gives a basis ## \{ x, x'_2, \ ... \ ... \ x'_{n 1} \}## of ##M## and it follows that ## \{ x, x'_2, \ ... \ ... \ x'_{n  1}, x'_n \}## is a basis of ##F## that contains ##x## ... ... "
My question is as follows:
Why/how exactly does it follow that ##\{ x, x'_2, \ ... \ ... \ x'_{n  1}, x'_n \}## is a basis of ##F## that contains ##x##. ... ...
Help will be appreciated ...
Peter
====================================================================================================
It may help PFmembers 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:
Hope that helps ... ...
Peter
Attachments

Bland  1  Proposition 4.3.14 ... ... PART 1.png82.6 KB · Views: 384

Bland  2  Proposition 4.3.14 ... ... PART 2.png57.9 KB · Views: 350

Bland  Definition 4.3.9 .png22.9 KB · Views: 405

?temp_hash=5442b546862ff55ca769a0f78124e05a.png82.6 KB · Views: 535

?temp_hash=5442b546862ff55ca769a0f78124e05a.png57.9 KB · Views: 312

?temp_hash=5442b546862ff55ca769a0f78124e05a.png22.9 KB · Views: 311