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?