Polynomial ring: Difference between revisions
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{{curing self-functor}} | |||
==Definition for commutative rings== | ==Definition for commutative rings== | ||
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The map sending a [[commutative unital ring]] to its polynomial ring is a self-functor on the category of commutative unital rings. | The map sending a [[commutative unital ring]] to its polynomial ring is a self-functor on the category of commutative unital rings. | ||
[[ | ==Related notions== | ||
* [[Multivariate polynomial ring]] | |||
* [[Laurent polynomial ring]] | |||
* [[Formal power series ring]] | |||
* [[Laurent series ring]] | |||
==Operations== | |||
===Completion=== | |||
The [[completion]] of the polynomial ring with respect to the [[ideal]] generated by the indeterminate, is the [[formal power series ring]]. | |||
===Localization and field of fractions=== | |||
* The [[localization]] of the polynomial ring at the multiplicative set comprising powers of <math>x</math>, is the [[Laurent polynomial ring]]. | |||
* The [[field of fractions]] of the [[polynomial ring over a field]] is the [[function field]], viz the field of rational functions over that field. The field of fractions ofthe polynomial ring over an integral domain, is the function field of its field of fractions. | |||
Revision as of 09:05, 8 August 2007
Definition for commutative rings
Definition with symbols
Let denote a commutative unital ring. The, the polynomial ring over in one variable, denoted as where is termed the indeterminate, is defined as the ring of formal polynomials in with coefficients in .
![{\displaystyle R[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440)
Functoriality
The map sending a commutative unital ring to its polynomial ring is a self-functor on the category of commutative unital rings.
Related notions
Operations
Completion
The completion of the polynomial ring with respect to the ideal generated by the indeterminate, is the formal power series ring.
Localization and field of fractions
- The localization of the polynomial ring at the multiplicative set comprising powers of , is the Laurent polynomial ring.
- The field of fractions of the polynomial ring over a field is the function field, viz the field of rational functions over that field. The field of fractions ofthe polynomial ring over an integral domain, is the function field of its field of fractions.