Understanding Bland's Proof of Proposition 4.3.14: Primitive Elements of Modules

[Math Amateur]

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 8326
View attachment 8327
In the above proof by Bland we read the following:

" ... ...If \(\displaystyle y \neq 0\), then we can write \(\displaystyle y = y' b\) where \(\displaystyle y'\) is a primitive element of \(\displaystyle F\) and \(\displaystyle b\) is a nonzero element of \(\displaystyle R\) ... ... "Can someone explain why/how it is that we can write \(\displaystyle y = y' b\) where \(\displaystyle y'\) is a primitive element of \(\displaystyle F\) and \(\displaystyle b\) is a nonzero element of \(\displaystyle R\) ... ... 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:View attachment 8328Hope that helps ...

Peter

 

[steenis]

This is a direct consequence of lemma 4.3.10 on page 123.