Polynomial ring

Definition for commutative rings

Definition with symbols

Let $R$ denote a commutative unital ring. The, the polynomial ring over $R$ in one variable, denoted as $R[x]$ where $x$ is termed the indeterminate, is defined as the ring of formal polynomials in $x$ with coefficients in $R$ .

Extra structure

The polynomial ring over any commutative unital ring $R$ is, first and foremost, a commutative unital ring. However, it has a number of additional structures, some of which are described below.

As an algebra over the original ring

The polynomial ring $R[x]$ is a $R$ -algebra. In fact, any $R$ -algebra generated by one element over $R$ , is a quotient, as a $R$ -algebra, of $R[x]$ .

As a graded ring

The polynomial ring comes with a natural gradation. The $d^{th}$ graded component of the polynomial ring is the $R$ -span of $x^{d}$ .

In fact, this makes $R[x]$ a connected graded $R$ -algebra.

As a filtered ring

The polynomial ring comes with a natural filtration. The $d^{th}$ filtered component of the polynomial ring is the subgroup comprising the polynomials of degree at most $d$ . This is the filtration corresponding to the gradation described above, and makes $R[x]$ a connected filtered $R$ -algebra.