Why Is the Containment xR ⊆ x1R Proper in Lemma 4.3.10?

In summary, the conversation discusses the proof of Lemma 4.3.10 in Paul E. Bland's book, "Rings and Their Modules". The proof involves constructing an increasing chain in a field F and showing that it must terminate due to the field being noetherian. The termination of the chain at x_n R implies that x_n is primitive. This is shown by extending the chain and reaching a contradiction.
  • #1
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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 to fully understand the proof of Lemma 4.3.10 ... ...

Lemma 4.3.10 and its proof read as follows:https://www.physicsforums.com/attachments/8283
View attachment 8284I do not follow the strategy of the proof ... for example Bland writes:

" ... ... Thus, \(\displaystyle xR \subseteq x_1 R\) and we claim that this containment is proper ... ... "But ... why is Bland proving that this containment is proper ... what is the point ... how is this furthering the proof ...?Peter
 
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  • #2
He wants to construct an increasing chain
$$xR \subsetneq x_1R \subsetneq x_2R \subsetneq \cdots $$

in $F$, in which each containment is proper.
But he clains that $F$ is noetherian (by prop.4.2.11) and therefore this chain must terminate.
 
  • #3
steenis said:
He wants to construct an increasing chain
$$xR \subsetneq x_1R \subsetneq x_2R \subsetneq \cdots $$

in $F$, in which each containment is proper.
But he clains that $F$ is noetherian (by prop.4.2.11) and therefore this chain must terminate.
Thanks for the help, Steenis ...

Appreciate your assistance ...

Just a further point ...

At the end of Bland's proof we read ...

" ... ... If the chain terminates at \(\displaystyle x_n R\) then \(\displaystyle x_n\) is primitive ... ... " Why/how does the chain terminating at \(\displaystyle x_n R\) imply that \(\displaystyle x_n\) is primitive ... ...?Peter***EDIT***

Thinking about it a bit more I suspect that if \(\displaystyle x_n\) is not primitive then we can extend the chain by showing \(\displaystyle x_n R \subsetneq x_{n+1} R \) ... but ... contradiction! ... the chain terminates at \(\displaystyle x_n\) ...

Is that correct ... ?

Peter
 
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  • #4
Yes correct. If $x_n$ is not primitive then $x_n = x_{n+1} b$ with $x_{n+1} \in F$ and $b \in R$ is not a unit. And then $x_n R \subsetneq x_{n+1} R$. The same argument is used a few times in the proof.
 

FAQ: Why Is the Containment xR ⊆ x1R Proper in Lemma 4.3.10?

1. What is a module over a principal ideal domain?

A module over a principal ideal domain is a vector space-like structure where the scalar values come from a principal ideal domain. This means that the scalar values are elements of a commutative ring where every ideal is generated by a single element.

2. What are primitive elements in a module over a principal ideal domain?

Primitive elements in a module over a principal ideal domain are elements that cannot be written as a non-zero multiple of another element in the module. They are the building blocks of the module and can be used to generate all other elements.

3. What is Lemma 4.3.10 in Bland's work on modules over principal ideal domains?

Lemma 4.3.10 in Bland's work is a theorem that states that every finitely generated module over a principal ideal domain can be decomposed into a direct sum of cyclic modules. This is a fundamental result in the study of modules over principal ideal domains.

4. How is Lemma 4.3.10 used in the study of modules over principal ideal domains?

Lemma 4.3.10 is used to simplify the study of modules over principal ideal domains by breaking down complex modules into simpler, cyclic modules. This decomposition allows for easier analysis and understanding of the properties of the module.

5. Can Lemma 4.3.10 be applied to modules over non-principal ideal domains?

No, Lemma 4.3.10 only applies to modules over principal ideal domains. Non-principal ideal domains have more complicated structure and do not have the same decomposition property as principal ideal domains.

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