TITLE: Looking for a Better Way to Think About Polynomial Rings
QUESTION [15 upvotes]: Given a commutative ring $R$, the polynomial ring in one variable $R[x]$ can be defined as the set of all the formal expressions $a_0+a_1x+\cdots+a_nx^n$ with 'obvious' rules of addition and multiplication.
What exactly do we mean by a 'variable' here is not very clear though.
My main question is the following:

Qustion 1. Does it mean anything to say that $ax=xa$ in $R[x]$ for all $a\in R$?

When we talk about multivariable polynomial rings, this approach becomes cumbersome.
Even when talking about $R[x, y]$, the multilpication seems rather artificial.
Further, we also have an isomorphism $R[x][y]\cong R[x, y]$.
This is making me a bit uncomfortable:

Question 2. In $R[x][y]$, it seems a bit odd to write $xy=yx$ (See Question 1) but we do certainly want to write this.

I know these questions are rather vague. So finally I can ask this: Is there a better way to think about polynomial rings? Also, can we intrinsically define what a variable is?
Thanks.

REPLY [3 votes]: I like the idea of using the concept of Monoid ring $R[M]$ where $R$ is a ring and $M$ is a monoid. The  elements of $R[M]$ are finite formal sums 
$$r_1m_1 + \cdots + r_p m_p $$
with the obvious addition and multiplication by using distributivity. 
If $M=\mathbb N$, the nonnegative natural numbers  under addition, we get the polynomial ring $R[x]$.  Here an element of $M$ is written $x^i$ with $x^ix^j = x^{i+j}$. 
But also if $M= \mathbb Z$, the integers under addition, and  working as above, we get the ring of Laurent polynomials. 
Note that we get polynomials in many variables using $R[x,y]= (R[x])[y]$.