# Exploring Paul E. Bland's "Rings and Their Modules" - Proposition 4.3.14

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• Math Amateur
In summary, the conversation discusses the book "Rings and Their Modules" by Paul E. Bland and the need for help in understanding Proposition 4.3.14. The proof for this proposition is provided and it is explained that if a certain element is a unit, then it can be used as a basis for the module. The conversation also includes a definition of "primitive element of a module" for further clarification. Ultimately, it is determined that $\{x_1, x_2, \dots, x_{n-1}, x\}$ is a basis for the module $F$.
Math Amateur
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MHB
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 ... ...

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View attachment 8324

In the above proof by Bland we read the following:

" ... ... If $$\displaystyle a_n \neq 0$$, let $$\displaystyle y = x_1 a_1 + x_2 a_2 + \ ... \ ... \ + x_{n-1} a_{n-1}$$, so that $$\displaystyle x = y + x_n a_n$$. If $$\displaystyle y = 0$$, then $$\displaystyle x = x_n a_n$$, so $$\displaystyle a_n$$ is a unit since $$\displaystyle x$$ is primitive. Thus $$\displaystyle \{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}$$ is a basis for $$\displaystyle F$$. ... ..."Can someone please explain exactly why/how $$\displaystyle \{ x_1, x_2, \ ... \ ... \ , x_{n-1}, x \}$$ is a basis for $$\displaystyle F$$ ... ...
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:
https://www.physicsforums.com/attachments/8325
Hope that helps ...

Peter

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Remember from linear algebra, that if
$$\{x_1, \cdots, x_i, \cdots, x_n \}$$
is a basis of F, then, if $c \neq 0$ then
$$\{x_1, \cdots, cx_i, \cdots, x_n \}$$
is also a basis of F.

Here we have that $x=x_n a_n$ and $a_n$ is a unit, thus $a_n \neq 0$ and if
$$\{x_1, \cdots, x_{n-1},x_n \}$$
is a basis of F, so
$$\{x_1, \cdots, x_{n-1}, x \}$$
is also a basis of F

steenis said:
Remember from linear algebra, that if
$$\{x_1, \cdots, x_i, \cdots, x_n \}$$
is a basis of F, then, if $c \neq 0$ then
$$\{x_1, \cdots, cx_i, \cdots, x_n \}$$
is also a basis of F.

Here we have that $x=x_n a_n$ and $a_n$ is a unit, thus $a_n \neq 0$ and if
$$\{x_1, \cdots, x_{n-1},x_n \}$$
is a basis of F, so
$$\{x_1, \cdots, x_{n-1}, x \}$$
is also a basis of F
Thanks Steenis ...

Peter

On second thoughts, my answer is nor correct, because R is not a field, but a commutative ring.

$\{x_1,x_2, \cdots, x_n \}$ is a basis for F, thus

$F=x_1R \oplus x_2R \oplus \cdots \oplus x_nR$

we have: $x=x_n a_n$ and $a_n$ is a unit

then $R=a_nR$ and $x_nR=xR$

thus $F=x_1R \oplus x_2R \oplus \cdots \oplus xR$

thus $\{x_1,x_2, \cdots, x \}$ is a basis for F

Last edited:
steenis said:
On second thoughts, my answer is nor correct, because R is not a field, but a commutative ring.

$\{x_1,x_2, \cdots, x_n \}$ is a basis for F, thus

$F=x_1R \oplus x_2R \oplus \cdots \oplus x_nR$

we have: $x=x_n a_n$ and $a_n$ is a unit

then $R=a_nR$ and $x_nR=xR$

thus $F=x_1R \oplus x_2R \oplus \cdots \oplus xR$

thus $\{x_1,x_2, \cdots, x \}$ is a basis for F

Thanks for clarifying that Steenis ...

Peter

## 1. What is the significance of Proposition 4.3.14 in Paul E. Bland's "Rings and Their Modules"?

Proposition 4.3.14 is an important result in the study of rings and modules, as it provides a clear understanding of the structure of finitely generated modules over a principal ideal domain (PID). It states that every finitely generated module over a PID is isomorphic to a direct sum of cyclic modules. This result has numerous applications in abstract algebra and serves as a fundamental tool in the classification and characterization of different types of modules.

## 2. How does Proposition 4.3.14 relate to other theorems in "Rings and Their Modules"?

Proposition 4.3.14 builds upon previous theorems and propositions in "Rings and Their Modules", such as the structure theorem for finitely generated modules over a PID. It provides a more specific and concrete understanding of the structure of these modules by decomposing them into direct sums of cyclic modules. Additionally, it highlights the importance and usefulness of PIDs in the study of modules and rings.

## 3. Can Proposition 4.3.14 be applied to modules over any type of ring?

No, Proposition 4.3.14 specifically applies to finitely generated modules over principal ideal domains. This is because PIDs have certain properties, such as unique factorization of elements, that allow for the decomposition of modules into direct sums of cyclic modules. This proposition does not hold for modules over other types of rings, such as non-principal ideal domains or fields.

## 4. How can Proposition 4.3.14 be used in practical applications?

Proposition 4.3.14 has various practical applications in fields such as coding theory, cryptography, and linear algebra. For example, it can be used to analyze and construct error-correcting codes, which have important applications in communication systems. It can also be used in cryptography to design secure encryption algorithms. In linear algebra, this proposition can be used to classify and analyze different types of vector spaces and their subspaces.

## 5. What are some potential extensions or generalizations of Proposition 4.3.14?

Proposition 4.3.14 can be extended to other types of rings, such as Euclidean domains or Dedekind domains. It can also be generalized to modules over other algebraic structures, such as algebras or groups. Additionally, there are variations and extensions of this proposition that consider different types of decompositions, such as primary decompositions or primary cyclic decompositions. These extensions and generalizations provide a deeper understanding of the structure of modules and their relationships with different types of rings and algebraic structures.

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