How do I define Polynomial ring $R[x]$ over a commutative ring $R$ with unity? 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..
 A: The ring $R[x]$ can be formally defined as follows.
The elements of $R[x]$ are all infinite sequences $(r_0,r_1,r_2,\dots)$ such that all but a finite number of the $r_i$ are $0$. The sum of two such sequences is defined in the natural way. 
The product is the convolution product. Let $(a_0,a_1,a_2,\dots)$ and $(b_0,b_1,b_2,\dots)$ be two such sequences. Their product is defined as $(c_0,c_1,c_2,\dots)$, where 
$$c_n=\sum_{i=0}^n a_i b_{n-i}.$$
It turns out that the object $(0,1,0,0,0,\dots)$ behaves like $x$ should. The polynomial ordinarily called $r_0+r_1x+r_2x^2+\cdots+ r_nx^n$ can be identified with the sequence $(r_0,r_1,\dots, r_n,0,0,0,\dots)$.  
A: One simple and natural definition of $\rm\,R[x]\,$ is the subring of linear maps on $\rm\,R^\Bbb N\,$ generated by the shift map $\rm\,x\!:\: (r_0,r_1,\ldots)\to (0,r_0,r_1,\ldots)\:$ and scalings $\rm\:r\!:\: (r_0,r_1,\ldots)\to (rr_0,rr_1,\ldots),\ r\in R.$  
Remark $\ $ Note that $\rm\,x\,$ is transcendental over the subring of scaling maps $\rm(\cong R),$ since
$$\rm\ (r_0 + r_1\, x + \cdots + r_n\, x^n)\, (1,0,0,\ldots)\ =\, (r_0,r_1,\ldots, r_n,0,0,\ldots)$$
which is nonzero when the polynomial is nonzero.
