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Re: Fine, Gaglione, Rosenburger, Introduction to Abstract Algebra, (2014), pp.377-9

Dear Dr. Fine,

I am trying to understand the formal description in the chapter on Polynomial Rings in Introduction to Abstract Algebra, and there is one particular part that is confusing me. Specifically, about 3/4 of the way down p.378 :

Now we let $\ x ,=, (,0,,1,,0,,...)$ and identify $(,r,,0,,0,,...)$ with $r \in R$ and then define $$\ x^0 ,=, (,1,,0,,...) ,=, \mathbb{1}$$

I don't understand what "identify" means in this context, especially when there is the repeated use of "and", making $r$ ambiguous w.r.t. $, x ,$ or $,x^0 ,$, especially when it says:

... if $\ f,=,(a_0,,a_1,,...,,,a_n,,0,,0,,...)$ it follows that $$rf = (ra_0,,ra_1,,...,,ra_n,,...,)$$

This is so because for all $, n \in \overline{\mathbb{N}} ,$ we have

$$rf(n),=\ r(0)f(n)\ +\ r(1)f(n-1)\ +,\cdots,+\ r(n,),f,(,0,)\ =\ ra_n$$

since $r,(i), =, 0$ for all $,i,>,0$

Please can you explain a bit what is meant over this section, especially with respect to $r$ ?