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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 \(\displaystyle a_n \neq 0\), let \(\displaystyle y = x_1 a_1 + x_2 a_2 + \ ... \ ... \ + x_{n-1} a_{n-1}\), so that \(\displaystyle x = y + x_n a_n\). If \(\displaystyle y = 0\), then \(\displaystyle x = x_n a_n\), so \(\displaystyle a_n\) is a unit since \(\displaystyle x\) is primitive. Thus \(\displaystyle \{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}\) is a basis for \(\displaystyle F\). ... ..."Can someone please explain exactly why/how \(\displaystyle \{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}\) is a basis for \(\displaystyle F\) ... ...

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 \(\displaystyle a_n \neq 0\), let \(\displaystyle y = x_1 a_1 + x_2 a_2 + \ ... \ ... \ + x_{n-1} a_{n-1}\), so that \(\displaystyle x = y + x_n a_n\). If \(\displaystyle y = 0\), then \(\displaystyle x = x_n a_n\), so \(\displaystyle a_n\) is a unit since \(\displaystyle x\) is primitive. Thus \(\displaystyle \{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}\) is a basis for \(\displaystyle F\). ... ..."Can someone please explain exactly why/how \(\displaystyle \{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}\) is a basis for \(\displaystyle F\) ... ...

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: