Polynomial ring - Revision history

Vipul: /* Related notions */

2012-07-04T01:55:05Z

Related notions

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	==Related notions==		==Related notions==

	~~For~~ a ~~complete list of variations of the notion~~ of ~~polynomial ring, refer:~~		===Variations that are the same over a field of characteristic zero===

	[[~~:Category:Variations~~ of polynomial ring]]		* [[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:		Some important notions include:

2012-07-04T01:35:12Z

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>.		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==		==Extra structure==

2009-02-05T16:59:59Z

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	==Related notions==		==Related notions==

			For a complete list of variations of the notion of polynomial ring, refer:

			[[:Category:Variations of polynomial ring]]

			Some important notions include:

	* [[Multivariate polynomial ring]]		* [[Multivariate polynomial ring]]

2009-02-05T16:59:24Z

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	~~{{polynomial-type ring construction}}~~

	==Definition for commutative rings==		==Definition for commutative rings==

2009-02-05T16:58:13Z

← Older revision		Revision as of 16:58, 5 February 2009
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			{{polynomial-type ring construction}}

	==Definition for commutative rings==		==Definition for commutative rings==

2008-05-12T16:28:32Z

6 revisions

← Older revision	Revision as of 16:28, 12 May 2008
(No difference)

2008-02-08T22:12:34Z

← Older revision		Revision as of 22:12, 8 February 2008
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	* 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 graded rings]]
	* A functor from the [[category of commutative unital rings]] to the [[category of filtered 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==		==Related notions==

2008-02-08T22:02:19Z

Extra structure

← Older revision		Revision as of 22:02, 8 February 2008
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	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.		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 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]]

	==Related notions==		==Related notions==

2008-02-08T22:01:12Z

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 ==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 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>.
 ==Related notions==
@@ Line 17: / Line 29: @@
 * [[Formal power series ring]]
 * [[Laurent series ring]]

2007-08-08T09:05:18Z

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 ==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.
-[[Category: Self-functors on commutative unital rings]]
+==Related notions==