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.

Functoriality

The polynomial ring can be viewed as a functor in any of the following senses:

• A functor from the category of commutative unital rings to itself
• A functor from the category of commutative unital rings to the category of graded rings
• A functor from the category of commutative unital rings to the category of filtered rings

Related notions

• Multivariate polynomial ring
• Laurent polynomial ring
• Formal power series ring
• Laurent series ring