Polynomial ring: Difference between revisions

Latest revision as of 01:55, 4 July 2012

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 $R$ .

Note that we could choose any letter for the indeterminate in place of $x$ and our notation would change accordingly. For instance, if we use the letter $t$ for the indeterminate, the ring is denoted $R[t]$ .

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.

Functoriality

The polynomial ring can be viewed as a functor in any of the following senses:

A functor from the category of commutative unital rings to itself
A functor from the category of commutative unital rings to the category of graded rings
A functor from the category of commutative unital rings to the category of filtered rings

Preservation of structure and properties

For many properties of commutative unital rings, it is true that when we pass to the polynomial ring, the property is preserved. However, this is not true for all properties of interest. For a list of properties that are preserved on passing to the polynomial ring, refer:

Category:Polynomial-closed properties of commutative unital rings

Related notions

Variations that are the same over a field of characteristic zero

Ring generated by binomial polynomials (makes sense for any commutative unital ring of characteristic zero)
Divided polynomial ring
Ring of integer-valued polynomials (makes sense over any integral domain)

Some important notions include:

@@ Line 3: / Line 3: @@
 ===Definition with symbols===
-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 on <math>R</math>.
+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>.
+Note that we could choose any letter for the indeterminate in place of <math>x</math> and our notation would change accordingly. For instance, if we use the letter <math>t</math> for the indeterminate, the ring is denoted <math>R[t]</math>.
+==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 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.
 ==Functoriality==
-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 can be viewed as a functor in any of the following senses:
+* A functor from the [[category of commutative unital rings]] to itself
+* A functor from the [[category of commutative unital rings]] to the [[category of graded rings]]
+* A functor from the [[category of commutative unital rings]] to the [[category of filtered rings]]
+==Preservation of structure and properties==
+For many [[property of commutative unital rings|properties of commutative unital rings]], it is true that when we pass to the polynomial ring, the property is preserved. However, this is not true for all properties of interest. For a list of properties that are preserved on passing to the polynomial ring, refer:
+[[:Category:Polynomial-closed properties of commutative unital rings]]
+==Related notions==
+===Variations that are the same over a field of characteristic zero===
+* [[Ring generated by binomial polynomials]] (makes sense for any [[commutative unital ring]] of characteristic zero)
+* [[Divided polynomial ring]]
+* [[Ring of integer-valued polynomials]] (makes sense over any [[integral domain]])
+Some important notions include:
-[[Category: Self-functors on commutative unital rings]]
+* [[Multivariate polynomial ring]]
+* [[Laurent polynomial ring]]
+* [[Formal power series ring]]
+* [[Laurent series ring]]