Multivariate polynomial ring

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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