The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials

Let $R$ be a commutative ring (Commutative ring) and let $I$ be a set. The set $R[x_i:i\in I]$ consists of the functions

$c:\mathcal{M}(I)⟶R$

with finite support, where $\mathcal{M}(I)$ is the monoid of Monomials on an index set as finitely supported exponent families. We write such a function as the formal sum ${\sum }_a{c}_a{x}^a$. Addition is pointwise. Multiplication is the convolution

$(cd{)}_u:={\sum }_{a+b=u}{c}_a{d}_b,$

where only pairs in $\mathrm{supp}(c)\times \mathrm{supp}(d)$ contribute and the sum is the finite sum of A finite sum in a commutative monoid indexed by an arbitrary finite set. The constant $r\in R$ is the coefficient family supported at the zero monomial with value $r$, and the indeterminate $x_i$ is supported at the exponent family that is $1$ at $i$ and $0$ elsewhere.

The convolution is well defined and these operations make the displayed set a commutative ring by Finite convolution makes $R[x_i:i\in I]$ a commutative ring containing $R$ ↗.