MathGloss

Let $R$ be a commutative ring and $x$ a variable. Then $R[x]$ is the polynomial ring of polynomials in $x$ with coefficients in $R$.

We can also define the polynomial ring in multiple variables. Let $X = {x_1,\dots,x_n}$ be a finite collection of variables. Then $R[X] = R[x_1,\dots,x_n]$ is defined as follows:

To each function $I:X\to \mathbb N$, associate a monomial $$x^I = x_1^{I(x_1)}+\cdots + x_n^{I(x_n)}.$$ Then $$R[X]= \left\{ \sum_{I:X\to \mathbb N} a_IX^I \mid a_I\in R, a_I = 0 \text{ for all but finitely many } I \right\}.$$ Multiplication and addition are defined exactly how you would think.

Wikidata ID: Q1455652

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