External and Internal Direct Sums - Bland - Rings and Their Modules

[Math Amateur]

In Paul E. Bland's book: Rings and Their Modules, the author defines the external direct sum of a family of \(\displaystyle R\)-modules as follows:
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Two pages later, Bland defines the internal direct sum of a family of submodules of an \(\displaystyle R\)-module as follows:
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I note that in the definition of the external direct sum, Bland defines the elements \(\displaystyle ( x_\alpha)\) of the external direct sum as follows:

\(\displaystyle ( x_\alpha) \in \prod_\Delta M_\alpha \ | \ ( x_\alpha) = 0 \text{ for almost all } \alpha \in \Delta \)

In doing this, the way I interpret him, Bland seems concerned that 'sums' of the form \(\displaystyle \sum_\alpha x_\alpha\) do not diverge ... but why? ... he has no such sums involved ...?

Can someone help clarify this issue?Further, however, when we inspect Bland's definition of the internal direct sum (see above), we find no such care with the definition of the \(\displaystyle x_\alpha\) - that is no mention of "\(\displaystyle ( x_\alpha) = 0 \text{ for almost all } \alpha \in \Delta\)"

... ... BUT ... ... with the internal direct sum as defined by Bland, we are definitely dealing with the sum

\(\displaystyle \sum_\Delta M_\alpha = \{ \sum_\Delta x_\alpha \ | \ x_\alpha \in M_\alpha \}\)Now, how do we know that these sums exist - that is, do not diverge ... ?


Well, my thinking is that since we are dealing with submodules \(\displaystyle M_\alpha\) of an \(\displaystyle R\)-module \(\displaystyle M\), each sum must 'converge' to an element \(\displaystyle x = \sum_\Delta x_\alpha\) ... ... ... whereas in the case of the external direct sum, no such guarantee exists ... ... Can someone please confirm/critique my thinking ... Help will be appreciated ... ...

Peter
***EDIT*** ... ... CONFUSION ... ...

Just a note to say I am now somewhat confused ... ...

I was checking the ideas of direct sums in D. G. Northcott's book: Lessons on Rings, Modules and Multiplicities and found the following introduction to direct sums including a definition of an internal direct sum ... ... as follows:https://www.physicsforums.com/attachments/3464Note that in his definition of internal direct sum above, unlike Bland, Northcott specifies "\(\displaystyle x_i = 0 \text{ for almost all i}\)" ... ...

... ... BUT ... ... maybe the difference is that Northcott talks about "each element \(\displaystyle x\) of \(\displaystyle M\)" having "a unique representation ... ... " ... ... maybe this explains the difference ...

Can someone please clarify the difference between Bland's and Nothcott's definitions ...

Peter

 

[Euge]

If you turn to Chapter 0, Section 0.4, you will see a discussion on the meaning of the notation $\sum_{\Delta} x_\alpha$. On page 5, Bland writes,

"From this point forward, all such sums $\sum_{\Delta} x_\alpha$ are to be viewed as finite sums and the expression '$x_{\alpha} = 0$ for almost all $\alpha \in \Delta$' will be omitted unless required for clarity."

 

[Math Amateur]

If you turn to Chapter 0, Section 0.4, you will see a discussion on the meaning of the notation $\sum_{\Delta} x_\alpha$. On page 5, Bland writes,

"From this point forward, all such sums $\sum_{\Delta} x_\alpha$ are to be viewed as finite sums and the expression '$x_{\alpha} = 0$ for almost all $\alpha \in \Delta$' will be omitted unless required for clarity."

Thanks Euge ... Oh ... Missed that ... Thanks ... That definitely clarifies the issue ... must read more carefully ...

Thanks again,

Peter