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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:

where

and where

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.

Peter

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.