I have struggled to define "Polynomial Ring" today. Since I'm not familiar with abstract algebra, i don't know if there is a theorem states that "For every commutative ring $R$ with unity, there exists a topology on $R$".

I'm wondering this, because i think, to directly define $R[x]$, $\sum\_{k=0}^\infty a\_k x^k$ should be defined first for arbitrary sequence $a$, that is, limit should be defined first. 

To avoid this, i first defined binary operations on a set $I$ of sequences $\sigma$ in $R$ such that $\sigma^{-1}(R\setminus \{0\})$ is finite. (In the usual way) Then I showed $I$ is a commutative ring with unity.

Then define a homomorphism to define $R[x]$. (That is, define $f(a)=\sum\_{k=0}^n a\_k x^k$ such that $n=\max\{i\in\omega:a\_i \neq 0\}$ for all $a\in I$ and let $R[x]\triangleq f(I)$).

Is my approach O.K? Or if there is a nice way to define $R[x]$, please let me know..