polynomial ring

plural polynomial rings

1. algebraA 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.

   Because they possess a rich theory and provide an excellent source of examples and counterexamples, integer-valued polynomial rings have attained some prominence in the theory of non-Noetherian commutative rings.

   2009, Jesse Elliott, “Some new approaches to integer-valued polynomial rings”, in Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson, editors, Commutative Algebra and Its Applications: Proceedings of the 5th International Fez Conference, Walter de Gruyter, page 223

   2 more examples

   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 R#91;x#93; is again right Noetherian.

   2000, Paul M. Cohn, Introduction to Ring Theory, Springer, page 106

   Synonyms: polynomial algebra, ring of polynomials