Ring based on families of integers - The Function Ring
I am trying to get a full understanding of Cohn Example 4 on page 9 - see attachment.
A formal and rigorous example showing clearly the nature and role of the families (and their indicies) involved would help enormously.
Cohn, page 9, Example 4 states:
"Let I be any set and denote by the set of all functions from I to Z.
Thus the elements of are families of integers indexed by I.
Addition and multiplication are defined componentwise:
It is very easily verified becomes a ring."
In order to understand this example fully, I would very much appreciate an example of what is at work here showing clearly the nature of the families and their indices.
Re: Ring based on families of integers - The Function Ring
Basically it fundamentally comes to the fact that if is the set of all functions from then should be countable. Namely.
there exists a map from .
adding 2 functions becomes adding their evaluation point wise. for all .
basically we can take the element in we attached a label "1" to and evaluate it there for both 2 functions, label "2" and evaluate it there,... for all natural numbers.
Again, we can do multiplication pointwise as well.