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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, ..., Xn, is the set of all polynomials
in those variables with coefficients in R. This ring is denoted R[X1, ..., Xn]. 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, ..., Xn].
- If R is an integral domain, so is R[X1, ..., Xn].
- If R is NoetherianRing theory In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. Introduction Rings of polynomials over fields have many special properties; properties that follow from the fact that polynomial rings are, then R[X1, ..., Xn] is Noetherian. This is the Hilbert basis theorem.
Commutative algebra Ring theoryIn mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. Please refer to the glossary of ring theory for the definitions of te Polynomials
