Sum of Submodules - infinite family case

[Math Amateur]

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

I am trying to understand Section 1.4 which introduces modules.

I need help with one of the definitions included in the statement of Proposition 1.4.4.

Proposition 1.4.4 reads as follows:

View attachment 3648

In (2) in the above Proposition Bland implicitly defines the sum of a family of submodules as follows:

\(\displaystyle \sum_{\Delta} M_\alpha \ = \ \{ \sum_{\Delta} x_\alpha \ | \ x_\alpha \in M_\alpha \text{ for all } \alpha \in \Delta \}\)Now the definition leaves open the possibility that the family \(\displaystyle \Delta\) is infinite, so shouldn't the definition include the statement:

" ... ... where \(\displaystyle x_\alpha = 0\) for almost all \(\displaystyle \alpha \in \Delta\) ... ... "... ... so as to effectively ensure that when the family \(\displaystyle \Delta\) is infinite that each sum \(\displaystyle \sum_{\Delta} x_\alpha\) is meaningful ... ...?

Can someone please help with this matter?

Peter

 

[Euge]

Hi Peter,

We've discussed this before, but those sums $\sum_{\Delta} x_{\alpha}$ are assumed to have all but finitely many terms equal to $0$. I don't have the book on me but the meaning of the notation should be explained somewhere in Chapter 0.

 

[Math Amateur]

Hi Peter,

We've discussed this before, but those sums $\sum_{\Delta} x_{\alpha}$ are assumed to have all but finitely many terms equal to $0$. I don't have the book on me but the meaning of the notation should be explained somewhere in Chapter 0.

Thanks Euge ... yes, forgot that we had discussed the issue ... indeed, just checked and saw the explanation on page 5 ...

Occasionally Bland does explicitly mention that "\(\displaystyle x_\alpha = 0\) for almost all \(\displaystyle \alpha \in \Delta\)" ... such as on page 43 when Bland defines an external direct sum .. ... which put me off guard a bit ...

Thanks again for your help ...

Peter