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 $R$.

Functoriality

The map sending a commutative unital ring to its polynomial ring is a self-functor on the category of commutative unital rings.

Related notions

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

Operations

Completion

The completion of the polynomial ring with respect to the ideal generated by the indeterminate, is the formal power series ring.

Localization and field of fractions

• The localization of the polynomial ring at the multiplicative set comprising powers of $x$, is the Laurent polynomial ring.
• The field of fractions of the polynomial ring over a field is the function field, viz the field of rational functions over that field. The field of fractions ofthe polynomial ring over an integral domain, is the function field of its field of fractions.