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

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In summary, the conversation discusses a proof in Paul E. Bland's book "Rings and Their Modules" in Section 4.3 about modules over principal ideal domains. The proof involves a proposition which states that a basis of a free module can contain a primitive element. The conversation also includes a request for help in understanding the proof and a discussion about the definition of a primitive element. The summary concludes with a thank you to another member for their assistance.
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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:View attachment 8318
View attachment 8319
In the above proof by Bland we read the following:

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

Why/how exactly does it follow that $$\displaystyle \{ x, x'_2, \ ... \ ... \ x'_{n - 1}, x'_n \}$$ is a basis of $$\displaystyle F$$ that contains $$\displaystyle x$$. ... ... Help will be 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 8322

Hope that helps ...

Peter

Last edited:
$F$ is a free module, $rank(F) = n$, $x$ is a primitive element of $F$, and suppose the induction hypothesis is true for free modules with $rank < n$

If $\{ x_1, x_2, \cdots, x_n \}$ is a basis for $F$ ($rank(F) = n$)

then $x = x_1 a_1 + x_2 a_2 + \cdots + x_n a_n$ for some $a_i \in R$, $i=1, 2, \cdots, n$

Suppose $a_n = 0$

Then $x \in M = x_1R \oplus x_2R \oplus \cdots \oplus x_{n-1}R$ (definition)

Notice that $F = M \oplus x_nR$

$M$ is a free module and $rank(M) = n-1 < n$

By the induction hypothesis, $M$ has a basis $\{ x, x’_2, \cdots, x’_{n-1} \}$, containing $x$

Thus $M = xR \oplus x’_2R \oplus \cdots \oplus x’_{n-1}R$

and $F = xR \oplus x’_2R \oplus \cdots \oplus x’_{n-1}R \oplus x_nR$

Thus $\{ x, x’_2, \cdots, x’_{n-1}, x_n \}$ is a basis of $F$ that contains $x$

steenis said:
$F$ is a free module, $rank(F) = n$, $x$ is a primitive element of $F$, and suppose the induction hypothesis is true for free modules with $rank < n$

If $\{ x_1, x_2, \cdots, x_n \}$ is a basis for $F$ ($rank(F) = n$)

then $x = x_1 a_1 + x_2 a_2 + \cdots + x_n a_n$ for some $a_i \in R$, $i=1, 2, \cdots, n$

Suppose $a_n = 0$

Then $x \in M = x_1R \oplus x_2R \oplus \cdots \oplus x_{n-1}R$ (definition)

Notice that $F = M \oplus x_nR$

$M$ is a free module and $rank(M) = n-1 < n$

By the induction hypothesis, $M$ has a basis $\{ x, x’_2, \cdots, x’_{n-1} \}$, containing $x$

Thus $M = xR \oplus x’_2R \oplus \cdots \oplus x’_{n-1}R$

and $F = xR \oplus x’_2R \oplus \cdots \oplus x’_{n-1}R \oplus x_nR$

Thus $\{ x, x’_2, \cdots, x’_{n-1}, x_n \}$ is a basis of $F$ that contains $x$

Thanks Steenis ...

Appreciate your help ...

Peter

## 1. What are primitive elements in a free module?

Primitive elements in a free module are those elements that cannot be expressed as a linear combination of other elements in the module. In other words, they are independent and cannot be reduced any further.

## 2. How are primitive elements related to basis elements in a free module?

In a free module, the basis elements are a set of primitive elements that generate the entire module. This means that any element in the module can be expressed as a linear combination of the basis elements.

## 3. Can a free module have more than one basis?

Yes, a free module can have multiple bases. This is because there can be different sets of primitive elements that can generate the same module. However, all bases for a given free module will have the same number of elements.

## 4. How can primitive elements be identified in a free module?

One way to identify primitive elements in a free module is by using the rank-nullity theorem. This theorem states that the rank of a free module is equal to the number of basis elements, so the primitive elements can be found by determining the basis elements.

## 5. What is the importance of primitive elements in free modules?

Primitive elements play a crucial role in the study of free modules. They help us understand the structure and properties of free modules, and they are essential in applications such as linear algebra and coding theory.

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