Understanding Lemma 4.3.12 in Paul E. Bland's Book: "Rings and Their 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 some further help in order to fully understand the proof of Lemma 4.3.12 ... ...

Lemma 4.3.12 reads as follows:View attachment 8316My question is as follows:

In the Lemma R is a PID ... where in the proof are the special properties of a PID used/needed ... it seems to me that the argument of the proof would hold valid for an ordinary commutative ring with identity ...

Can someone please point out the points in the proof where the special properties of a PID are needed ...

Peter

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It may help MHB
readers of the above post to have access to Bland's definition of a primitive element ... so I am providing the same as follows:View attachment 8317
Hope that helps ...

Peter

 

[steenis]

The answer to your question is on page 120 of Bland, quote: "A greatest common divisor of a set of nonzero elements of R may fail to exist. The case is different for a principal ideal ring."
So, if you want existing gcd's, your ring has to be a PID.

 

[Math Amateur]

The answer to your question is on page 120 of Bland, quote: "A greatest common divisor of a set of nonzero elements of R may fail to exist. The case is different for a principal ideal ring."
So, if you want existing gcd's, your ring has to be a PID.

Thanks Steenis ...

I appreciate your help ...

Peter