polynomial ringとは 意味・読み方・使い方

名詞

polynomial ring (複数形 polynomial rings)

1. (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 ${\displaystyle A}$ is a graded ring over a local ring, ${\displaystyle A}$ 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 ${\displaystyle R}$ is any right Noetherian ring, the polynomial ring ${\displaystyle R[x]}$ is again right Noetherian.

   • 2009, Jesse Elliott, Some new approaches to integer-valued polynomial rings, 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.