Meaning of polynomial ring | Babel Free

Definitions

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.

Equivalents

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⟳.”

“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.”

“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.”

CEFR level