 #1
Math Amateur
Gold Member
MHB
 3,990
 48
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:
" ... ... 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:
Peter
Attachments

Bland  1  Proposition 4.3.14 ... ... PART 1.png49.6 KB · Views: 425

Bland  2  Proposition 4.3.14 ... ... PART 2.png35.9 KB · Views: 392

Bland  Definition 4.3.9 .png13.3 KB · Views: 450

?temp_hash=5442b546862ff55ca769a0f78124e05a.png49.6 KB · Views: 614

?temp_hash=5442b546862ff55ca769a0f78124e05a.png35.9 KB · Views: 406

?temp_hash=5442b546862ff55ca769a0f78124e05a.png13.3 KB · Views: 379