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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:View attachment 8318

View attachment 8319

In the above proof by Bland we read the following:

" ... ... The induction hypothesis gives a basis \(\displaystyle \{ x, x'_2, \ ... \ ... \ x'_{n -1} \}\) of \(\displaystyle M\) and it follows that \(\displaystyle \{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}\) is a basis of \(\displaystyle F\) that contains \(\displaystyle x\). ... ... "My question is as follows:

Why/how exactly does it follow that \(\displaystyle \{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}\) is a basis of \(\displaystyle F\) that contains \(\displaystyle x\). ... ... Help will be 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:

View attachment 8322

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:View attachment 8318

View attachment 8319

In the above proof by Bland we read the following:

" ... ... The induction hypothesis gives a basis \(\displaystyle \{ x, x'_2, \ ... \ ... \ x'_{n -1} \}\) of \(\displaystyle M\) and it follows that \(\displaystyle \{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}\) is a basis of \(\displaystyle F\) that contains \(\displaystyle x\). ... ... "My question is as follows:

Why/how exactly does it follow that \(\displaystyle \{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}\) is a basis of \(\displaystyle F\) that contains \(\displaystyle x\). ... ... Help will be 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:

View attachment 8322

Hope that helps ...

Peter

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