How do you interpret the indeterminate "x" in ring theory from the set theory viewpoint? How do you write down R[x] as a set? Is it appropriate/correct to just say that

$R[x] = \{ f: R \to R | \exists n \in \mathbb{N}, \mbox{ and } c_0, \dots, c_n \in R \mbox{ such that } f(x) = c_0 + c_1 x + \dots + c_n x^n \}$

This appears to be a very analytic definition. Is there a better definition that highlights the algebraic aspect of the set of polynomials? $$ $$ EDIT: There is a thread on this at http://meta.mathoverflow.net/discussion/568/inconsistent-and-closedminded-question-closing/#Item_0 There is a single user who is composing a lengthy response to this question. Anyway, take a look.

How do you interpret the indeterminate "x" in ring theory from the set theory viewpoint? How do you write down R[x] as a set? Is it appropriate/correct to just say that

$R[x] = \{ f: R \to R | \exists n \in \mathbb{N}, \mbox{ and } c_0, \dots, c_n \in R \mbox{ such that } f(x) = c_0 + c_1 x + \dots + c_n x^n \}$

This appears to be a very analytic definition. Is there a better definition that highlights the algebraic aspect of the set of polynomials?