Multivariate polynomial ring

This is a variation of polynomial ring
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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