Definition of polynomial ring Given a ring $R$, the polynomial ring is defined as
$$R [x]:=\left\{\sum_{k=0}^n a_k x^k:  n\ge0, \ a_k \in R \ \text{for }k\in \left\{0, 1, \dots, n \right\}\right\}. $$
However, it is not usually specified what $x$ is. In order for multiplication to make sense, I guess it has to be an element in $R$ at least. But is $R[x]$ the set of all functions $ P: R\rightarrow R, \text{given by } x\mapsto \sum_{k=0}^n a_k x^k$, or the set of those functions evaluated at $x$?
Sometimes $R$ is required to be commutative. Does that make any difference for R[x]?
 A: We think of polynomials as infinite lists which have a finite point in after which everything is $0$. (Meaning that a polynomial is of finite degree by definition.)
For example the polynomial $x^2+1$ is actually the infinite list $(1,0,1,0,0,...)$. (Think of the first element as the $0$th element, then $n$th element in the list corresponds to the coefficient of $x^n$, notice that the list can contain numbers or more generally, elements of any ring.)
The operations are defined on this concept but this writing is not practical so we switch to the notation with $x$.
Usually the books just say $x$ to be an indeterminate.
A: Some hints:
First, $x$ is an unknown over $R$. This means that $x$ does not satisfy any nontrivial equation over $R$:
$\sum_{k=0}^n a_kx^k=0$ with not all $a_k$ vanishing.
Second, consider the set of all functions $P:x\mapsto \sum_{k=0}^n a_kx^k$. If $R$ is an infinite field of char 0, two different polynomials define different polynomial functions, but this property is false for finite fields.
Third, if $R$ is commutative, then $R[x]$ is commutative.
