Editing Polynomial ring (section)

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In [[mathematics]], especially in the field of [[algebra]], a '''polynomial ring''' or '''polynomial algebra''' is a [[ring (mathematics)|ring]] formed from the [[set (mathematics)|set]] of [[polynomial]]s in one or more [[indeterminate (variable)|indeterminate]]s (traditionally also called [[variable (mathematics)|variables]]) with [[coefficient]]s in another [[ring (mathematics)|ring]], often a [[field (mathematics)|field]].

Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the [[Integer#Algebraic_properties|integers]].

Polynomial rings occur and are often fundamental in many parts of mathematics such as [[number theory]], [[commutative algebra]], and [[algebraic geometry]]. In [[ring theory]], many classes of rings, such as [[unique factorization domain]]s, [[regular ring]]s, [[group ring]]s, [[formal power series|rings of formal power series]], [[Ore polynomial]]s, [[graded ring]]s, have been introduced for generalizing some properties of polynomial rings.

A closely related notion is that of the [[ring of polynomial functions]] on a [[vector space]], and, more generally, [[ring of regular functions]] on an [[algebraic variety]].