Polynomials in several variables

Definition

Let $R$ be a commutative ring and let $n \ge 1$ be an integer. For $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n$, the monomial $x^\alpha$ is the formal product $x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. A polynomial in $x_1, \dots, x_n$ over $R$ is a finite sum $\sum_\alpha c_\alpha x^\alpha$ with $c_\alpha \in R$. Two polynomials are equal if and only if they have the same coefficient $c_\alpha$ for each $\alpha$. The ring $R[x_1, \dots, x_n]$ is the set of these polynomials, with coefficientwise addition and the product $$\Big(\sum_\alpha c_\alpha x^\alpha\Big) \Big(\sum_\beta d_\beta x^\beta\Big) = \sum_\gamma \Big(\sum_{\alpha + \beta = \gamma} c_\alpha d_\beta\Big) x^\gamma.$$