polynomial ring

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polynomial ring

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Noun edit

polynomial ring (plural polynomial rings)

(algebra) 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.
- 1998, Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, page 270:
  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.
- 2000, Paul M. Cohn, Introduction to Ring Theory, Springer, page 106:
  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 $R[x]$ is again right Noetherian.
- 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:
  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.

Usage notes edit

$K[X]$ is called the polynomial ring (or ring of polynomials) in $X$ over $K$ .
The notation $K[X_{1},X_{2},\dots ,X_{n}]$ is standardly used for the case of a polynomial ring in multiple variables.
- It is possible to speak of a polynomial ring in an infinite set of variables, provided it is assumed that any individual polynomial depends only on a finite number of variables.