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

In summary, the proof in Proposition 4.3.14 by Paul E. Bland states that if y is not equal to 0, then it can be written as y' multiplied by b, where y' is a primitive element of F and b is a nonzero element of R. This can be further explained by referencing Bland's definition of 'primitive element of a module'.
  • #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 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
 
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This is a direct consequence of lemma 4.3.10 on page 123.
 

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

What is Bland's Proof of Proposition 4.3.14?

Bland's Proof of Proposition 4.3.14 is a mathematical proof that provides a method for finding the primitive elements of a module. These elements are important in understanding the structure and properties of modules.

Why is Proposition 4.3.14 important?

Proposition 4.3.14 is important because it helps us to understand the structure of modules and their elements. It also provides a method for finding primitive elements, which can be useful in solving problems and proving other theorems related to modules.

What are primitive elements of a module?

Primitive elements of a module are elements that cannot be expressed as a product of other elements in the module. They are essentially the building blocks of the module and are important in understanding its structure and properties.

How is Bland's Proof of Proposition 4.3.14 used in practice?

Bland's Proof of Proposition 4.3.14 is used in practice to find primitive elements of modules. This can be useful in solving problems in various branches of mathematics, such as algebra and number theory.

What are some examples of applications of Bland's Proof of Proposition 4.3.14?

Some examples of applications of Bland's Proof of Proposition 4.3.14 include finding primitive elements in vector spaces, solving linear Diophantine equations, and proving other theorems related to modules and their elements.

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