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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 4.3.12 ... ...
Proposition 4.3.12 reads as follows:View attachment 8314In the above proof by Bland we read the following:
" ... ... Since $$x$$ is primitive, $$d$$ is a unit, so $$d$$ and $$1$$ are associates. Thus $$1$$ is a greatest common denominator of $$\{ a_\alpha \ \mid \ a_\alpha \neq 0 \}$$. ... ... "Can someone please explain exactly why $$d$$ and $$1$$ being associates implies that $$1$$ is a greatest common denominator of $$\{ a_\alpha \ \mid \ a_\alpha \neq 0 \}$$ ... ...Peter==============================================================================
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 8315Hope 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 4.3.12 ... ...
Proposition 4.3.12 reads as follows:View attachment 8314In the above proof by Bland we read the following:
" ... ... Since $$x$$ is primitive, $$d$$ is a unit, so $$d$$ and $$1$$ are associates. Thus $$1$$ is a greatest common denominator of $$\{ a_\alpha \ \mid \ a_\alpha \neq 0 \}$$. ... ... "Can someone please explain exactly why $$d$$ and $$1$$ being associates implies that $$1$$ is a greatest common denominator of $$\{ a_\alpha \ \mid \ a_\alpha \neq 0 \}$$ ... ...Peter==============================================================================
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 8315Hope that helps ...
Peter
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