The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Let $R$ be a commutative ring (Commutative ring). A function $a\text{\hspace{0.17em}⁣}:\mathbb{N}\to R$ has finite support when there is $N\in \mathbb{N}$ such that $a_i=0$ for every $i>N$. The polynomial ring $R[x]$ is the set of all finitely supported functions $a\text{\hspace{0.17em}⁣}:\mathbb{N}\to R$.

For $a,b\in R[x]$ and $n\in \mathbb{N}$, define

$(a+b{)}_n:=a_n+b_n,(ab{)}_n:={\sum }_{i+j=n}{a}_i{b}_j\mathrm{.}$

The convolution sum is a finite sum in the additive commutative monoid of $R$ (A finite sum in a commutative monoid indexed by an arbitrary finite set). Write $0$ for the zero sequence, $1$ for the sequence with coefficient $1_R$ at index $0$ and zero elsewhere, and $x$ for the sequence with coefficient $1_R$ at index $1$ and zero elsewhere. The coefficient sequence supported at $0$ with value $r$ is denoted again by $r$, and a polynomial $a$ is written formally as ${\sum }_i{a}_i{x}^i$.

The closure of these operations and the commutative-ring axioms are established by Coefficientwise sums and convolution products of finitely supported sequences are finitely supported ↗ and Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring ↗.