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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,X₁, ..., X_n, is the set of all polynomials in those variables with coefficients in R. This ring is denoted R[X₁, ..., X_n]. For example, an 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[X₁, ..., X_n].
• If R is an integral domain, so is R[X₁, ..., X_n].
• If R is Noetherian, then R[X₁, ..., X_n] is Noetherian. This is the Hilbert basis theorem.