Help Understanding Bland's Proposition 4.3.14 in Rings and Their Modules

In summary, the conversation discusses the proof of Proposition 4.3.14 from Paul E. Bland's book "Rings and Their Modules". The question is why it follows that xR = x_1R, and the answer is given through equations and explanations. It is shown that since x is primitive and x = x_1a, where a is a unit, it follows that xR = x_1R. Another way to understand this is by considering that any element in xR can also be written as x_1ar, where r is an element of R. Similarly, any element in x_1R can be written as x_1s, where s is an element of R. Therefore
  • #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 further help in order to fully understand the proof of Proposition 4.3.14 ... ...

Proposition 4.3.14 reads as follows:View attachment 8320
View attachment 8321In the above proof by Bland we read the following:

" ... ... If \(\displaystyle \{ x_1 \}\) is a basis for \(\displaystyle F\), then there is an \(\displaystyle a \in R\) such that \(\displaystyle x = x_1 a\). But \(\displaystyle x\) is primitive, so \(\displaystyle a\) is a unit in \(\displaystyle R\). Hence \(\displaystyle x R = x_1 R\) ... ... "

My question is as follows:

Why in the above quote, does it follow that \(\displaystyle x R = x_1 R\) ... ... ?Is it because \(\displaystyle x = x_1 a\) where \(\displaystyle a\) is a unit ... ... ... ... ... (1)

Hence \(\displaystyle xR = x_1 a R\) ... ... I presume this follows (1)

Therefore \(\displaystyle xR = x_1 (a R )\)

But \(\displaystyle aR = R\) since a is a unit ... ''

So \(\displaystyle xR = x_1 R \)

Is that correct?

Peter
 
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  • #2
Yes, you did it correct.

You can do it also this way:

Let $y \in xR$ then $y=xr=x_1ar \in x_1R$
and
Let $y \in x_1R$ then $y=x_1s=xa^{-1}s \in xR$
 
  • #3
steenis said:
Yes, you did it correct.

You can do it also this way:

Let $y \in xR$ then $y=xr=x_1ar \in x_1R$
and
Let $y \in x_1R$ then $y=x_1s=xa^{-1}s \in xR$

Thanks for the help, Steenis ...

Peter
 

FAQ: Help Understanding Bland's Proposition 4.3.14 in Rings and Their Modules

1. What is Bland's Proposition 4.3.14 in Rings and Their Modules?

Bland's Proposition 4.3.14 is a theorem in abstract algebra that deals with rings and their modules. It states that if a module M is finitely generated, then it is isomorphic to the direct sum of its cyclic submodules.

2. How is Bland's Proposition 4.3.14 useful in ring theory?

This proposition is useful in proving other theorems and properties related to rings and their modules. It also helps in simplifying calculations and understanding the structure of modules.

3. Can you explain the concept of direct sum in relation to Bland's Proposition 4.3.14?

Direct sum is a way of combining two or more modules to form a new module. In Bland's Proposition 4.3.14, it is used to show that a finitely generated module can be broken down into simpler cyclic submodules.

4. What are some real-life applications of Bland's Proposition 4.3.14?

Bland's Proposition 4.3.14 is often used in algebraic geometry, representation theory, and number theory. It also has applications in coding theory and cryptography.

5. Are there any limitations to Bland's Proposition 4.3.14?

Yes, Bland's Proposition 4.3.14 only applies to finitely generated modules. It also does not hold for infinite direct sums or products of modules.

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