Free Modules w/Multiple Bases: 2nd Issue w/Example 5 (Paul E. Bland)

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am trying to understand Section 2.2 on free modules and need help with Example 5 showing a module with two bases ... ...

Thanks to Caffeinemachine, I have largely clarified one issue/problem I had with Example 5, but now have a second, separate issue ... (see below)Example 5 reads as follows:https://www.physicsforums.com/attachments/3577In an argument that begins: (see above text from Bland)

" ... ... So if \(\displaystyle R = \text{End}_{ \mathbb{Z} } (M)\), then ... ... "

Bland concludes that

" ... ... \(\displaystyle R\) has a basis with one element and a basis with two elements ... ... "

To say that I do not follow this argument would be an understatement!

Can someone help me to understand Bland's argument?

I would appreciate help in this matter ...

Peter

 

[Euge]

Hi Peter,

I'm guessing that statement was confusing since in a vector space setting, bases of the same vector space have the same cardinality. What Bland showed in his example is that an $R$-module may have two bases of different cardinality. For other examples, read the article in the following link:

example of free module with bases of diffrent cardinality | planetmath.org

He's also doing this example to motivate the concept of directly finite and infinite modules. Already, you know that finite dimensional vector spaces are directly finite modules. In Bland's example, however, $M$ is a directly infinite module.