Polynomial ring: Difference between revisions

Revision as of 22:01, 8 February 2008

Definition for commutative rings

Definition with symbols

Let $R$ denote a commutative unital ring. The, the polynomial ring over $R$ in one variable, denoted as $R[x]$ where $x$ is termed the indeterminate, is defined as the ring of formal polynomials in $x$ with coefficients in $R$ .

Extra structure

The polynomial ring over any commutative unital ring $R$ is, first and foremost, a commutative unital ring. However, it has a number of additional structures, some of which are described below.

As an algebra over the original ring

The polynomial ring $R[x]$ is a $R$ -algebra. In fact, any $R$ -algebra generated by one element over $R$ , is a quotient, as a $R$ -algebra, of $R[x]$ .

As a graded ring

The polynomial ring comes with a natural gradation. The $d^{th}$ graded component of the polynomial ring is the $R$ -span of $x^{d}$ .

In fact, this makes $R[x]$ a connected graded $R$ -algebra.

As a filtered ring

The polynomial ring comes with a natural filtration. The $d^{th}$ filtered component of the polynomial ring is the subgroup comprising the polynomials of degree at most $d$ . This is the filtration corresponding to the gradation described above, and makes $R[x]$ a connected filtered $R$ -algebra.

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-{{curing self-functor}}
 ==Definition for commutative rings==
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 Let <math>R</math> denote a [[commutative unital ring]]. The, the polynomial ring over <math>R</math> in one variable, denoted as <math>R[x]</math> where <math>x</math> is termed the ''indeterminate'', is defined as the ring of formal polynomials in <math>x</math> with coefficients in <math>R</math>.
-==Functoriality==
+==Extra structure==
+The polynomial ring over any commutative unital ring <math>R</math> is, first and foremost, a commutative unital ring. However, it has a number of additional structures, some of which are described below.
+===As an algebra over the original ring===
-The map sending a [[commutative unital ring]] to its polynomial ring is a self-functor on the category of commutative unital rings.
+The polynomial ring <math>R[x]</math> is a <math>R</math>-algebra. In fact, any <math>R</math>-algebra generated by one element over <math>R</math>, is a quotient, as a <math>R</math>-algebra, of <math>R[x]</math>.
+===As a graded ring===
+The polynomial ring comes with a natural ''gradation''. The <math>d^{th}</math> graded component of the polynomial ring is the <math>R</math>-span of <math>x^d</math>.
+In fact, this makes <math>R[x]</math> a connected graded <math>R</math>-algebra.
+===As a filtered ring===
+The polynomial ring comes with a natural ''filtration''. The <math>d^{th}</math> filtered component of the polynomial ring is the subgroup comprising the polynomials of degree at most <math>d</math>. This is the filtration corresponding to the gradation described above, and makes <math>R[x]</math> a connected filtered <math>R</math>-algebra.
 ==Related notions==
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 * [[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.