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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 $\displaystyle Z^I $ the set of all functions from I to Z.

Thus the elements of $\displaystyle Z^I $ are families $\displaystyle (x_i) $ of integers indexed by I.

Addition and multiplication are defined componentwise:

$\displaystyle (x_i) = (y_i) = (z_i) $ where $\displaystyle z_i = x_i + y_i $

and $\displaystyle (x_i) (y_i) = (t_i) $ where $\displaystyle t_i = x_i y_i$

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 $\displaystyle I$ is the set of all functions from $\displaystyle I \to \mathbb{Z}$ then $\displaystyle I$ should be countable. Namely.

there exists a map from $\displaystyle I \to \mathbb{N} $.

adding 2 functions becomes adding their evaluation point wise. $\displaystyle \bar{f} + f = \bar{f}(i_n) + f(i_n) $ for all $\displaystyle \mathbb{N} $.

basically we can take the element in $\displaystyle I $ 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.