MHB Infinite Direct Sums and Standard Inclusions and Projections

[Math Amateur]

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).

In Chapter2: Direct Sums and Short Exact Sequences in Sections 2.1.11 and 2.1.12 B&K deal with infinite direct products and infinite direct sums (external and internal).

In Section "2.1.13 Remarks", B&K comment on the implications of Sections 2.1.11 and 2.1.12.

Remark (ii) reads as follows:View attachment 3355My question related to B&K's remark above is as follows:

Why is it impossible to attach a meaning to an infinite set of maps (standard inclusions and projections that is ...)? We can attach meaning to an infinite set of modules or submodules ... ... Why is it more abstract or difficult (in fact, impossible!) to attach meaning to an infinite set of maps?

Could someone please clarify and explain B&Ks remark.

Peter***NOTE***

I am aware that in the text displayed above, B&K talk about an infinite 'sum' of maps saying ..." ... ... since it is impossible to attach a meaning to an infinite sum of maps ... ... "

BUT ... ... as far as I can see, the proposition to which they refer (Proposition 2.1.7) only seems to involve a list, not a sum of maps ...

Section 2.1.7 reads as follows:View attachment 3356

 

[Deveno]

There is no problem with assigning meaning to an infinite SET of maps.

But if we try to add the images of each map together, we have to wait an awfully long time for the "result".