Primitive Elements and Free Modules .... ....

In summary, Bland's book "Rings and Their Modules" discusses modules over principal ideal domains, specifically focusing on Section 4.3. In Proposition 4.3.14, Bland uses the induction hypothesis to prove that the set {x, x'_2, ..., x'_{n-1}, x'_n} is a basis for F that contains x. This is because the induction hypothesis states that {x, x'_2, ..., x'_{n-1}} is a basis for M, and therefore F can be expressed as the direct sum of M and x_nR. This result is important for understanding the definition of a 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 some help in order to fully understand the proof of Proposition 4.3.14 ... ...

Proposition 4.3.14 reads as follows:
?temp_hash=5442b546862ff55ca769a0f78124e05a.png

?temp_hash=5442b546862ff55ca769a0f78124e05a.png
In the above proof by Bland we read the following:

" ... ... The induction hypothesis gives a basis ## \{ x, x'_2, \ ... \ ... \ x'_{n -1} \}## of ##M## and it follows that ## \{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}## is a basis of ##F## that contains ##x## ... ... "My question is as follows:

Why/how exactly does it follow that ##\{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}## is a basis of ##F## that contains ##x##. ... ...Help will be appreciated ...

Peter====================================================================================================

It may help PFmembers 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:
?temp_hash=5442b546862ff55ca769a0f78124e05a.png
Hope that helps ... ...

Peter
 

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  • #2
That ##\{x_1,...,x_n\}## is a basis for ##F## means the same as that
$$F=x_1R\oplus...\oplus x_nR$$
and since ##M\triangleq x_1R\oplus...\oplus x_{n-1}R## we have ##F=M\oplus x_nR##.

Since the induction hypothesis gives us that ##\{x, x'_2,...,x'_{n-1}\}## is a basis of ##M## we have
$$M = xR\oplus x'_2R\oplus x'_3R\oplus...\oplus x'_{n-1}$$
So we have
$$F=M\oplus x_nR
= xR\oplus x'_2R\oplus x'_3R\oplus...\oplus x'_{n-1}\oplus x_nR$$
which is equivalent to saying that ##\{x,x'_2,...,x'_{n-1},x_n\}## is a basis for ##F##.
 
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  • #3
Thanks Andrew ...

Appreciate your help ...

Peter
 

1. What are primitive elements and free modules?

Primitive elements are elements of a ring or algebra that cannot be decomposed into simpler elements. Free modules are modules over a ring or algebra that have a basis, meaning they can be generated by linear combinations of a finite set of elements.

2. How do primitive elements and free modules relate to each other?

Primitive elements can be used to define a basis for a free module. This means that a free module can be built by combining primitive elements in different ways.

3. What is the significance of primitive elements and free modules in mathematics?

Primitive elements and free modules are important concepts in abstract algebra, specifically in the study of rings and modules. They provide a way to understand the structure and properties of these mathematical objects.

4. Can all elements of a ring or algebra be primitive?

No, not all elements can be primitive. In fact, most elements of a ring or algebra are not primitive. Primitive elements are a specific subset of elements that have certain properties and cannot be broken down into simpler elements.

5. Are there applications of primitive elements and free modules outside of mathematics?

Yes, these concepts have applications in various areas such as physics, computer science, and engineering. For example, they are used in quantum mechanics to study the behavior of particles, in computer graphics to create 3D models, and in signal processing to analyze and manipulate signals.

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