Multivariate polynomial ring

Definition
Let $$R$$ be a commutative unital ring. The $$n$$-variate polynomial ring over $$R$$ is defined as the ring of polynomials in $$n$$ symbols. If the $$n$$ symbols are $$x_1, x_2, \ldots, x_n$$, then the polynomial ring is $$x_1, x_2, \ldots, x_n$$.

The $$n$$-variate polynomial ring can be obtained by applying the polynomial ring operator $$n$$ times in succession.

When we simply say multivariate polynomial ring, we usually mean multivariate polynomial ring over a field.

We can also consider the polynomial ring in infinitely many variables over $$R$$.

Extra structure
The multivariate polynomial ring over a ring $$R$$ is, first and foremost, a commutative unital ring. However, it has a number of additional structures, as described below.

As an algebra over the original ring
The polynomial ring $$R[x_1,x_2,\ldots,x_n]$$ naturally gets the structure of a $$R$$-algebra. In fact it is free in the category of $$R$$-algebras, on $$n$$ generators.

A similar statement holds for polynomial rings in infinitely many variables.

As a graded ring
The polynomial ring $$R[x_1,x_2,\ldots,x_n]$$ naturally gets the structure of a connected graded $$R$$-algebra (and hence a graded ring). The $$d^{th}$$ graded component is the free $$R$$-module spanned by all monomials of total degree $$d$$.

The same holds when we have infinitely many variables.

As a filtered ring
The polynomial ring $$R[x_1,x_2,\ldots,x_n]$$ naturally gets the structure of a connected filtered $$R$$-algebra (and hence a filtered ring). The $$d^{th}$$ filtered component is the subgroup comprising polynomials of degree at most $$d$$.

Related notions

 * Polynomial ring
 * Laurent polynomial ring
 * Power series ring