polynomial ring

Noun

polynomial ring (plural polynomial rings)

1. A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K.
   • It then follows that if A is a graded ring over a local ring, A is a homomorphic image of a polynomial ring over a regular local ring. For the sake of brevity, we refer to a graded polynomial ring over a regular local ring simply as a graded polynomial ring.1998, Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, page 270
   • In Section 3.2 we shall study the special properties of a polynomial ring over a field; for the moment we note a property of polynomial rings which applies quite generally, the Hilbert basis theorem (after David Hilbert, 1862-1943):
     Theorem 3.3
     If R is any right Noetherian ring, the polynomial ring...2000, Paul M. Cohn, Introduction to Ring Theory, Springer, page 106

   Synonyms:

   polynomial algebra, ring of polynomials