# Polynomial ring

In abstract algebra, a**polynomial ring**is the set of polynomials in one or more variables with coefficients in a commutative ring.

More precisely, let *R* be a commutative ring. The polynomial ring in *n variables,*X_{1}*, ..., *X_{n}'', is the set of all polynomials
in those variables with coefficients in *R*. This ring is denoted *R*[*X _{1}*, ...,

*X*]. For example, an

_{n}**integer polynomial**is a polynomial with coefficients in the ring

*Z*of integers. This is something different from an integer-valued polynomial. Every commutative ring that is a finitely-generated algebra over a field can be written as a quotient of a polynomial ring.

The definition of a polynomial ring also works for noncommutative rings. The variables all commute with each other, and with each element of *R*. You can also define a ring where the variables do *not* commute with each other. This is known as the free algebra over *R*.

Polynomial rings are studied in the field of Commutative algebra.

## Properties

- If
*R*is a field, then*R*[*X*] is a principal ideal domain (and even a Euclidean domain). - If
*R*is a unique factorization domain, so is*R*[*X*, ...,_{1}*X*]._{n} - If
*R*is an integral domain, so is*R*[*X*, ...,_{1}*X*]._{n} - If
*R*is Noetherian, then*R*[*X*, ...,_{1}*X*] is Noetherian. This is the Hilbert basis theorem._{n}