Homogeneous ideal - Revision history

Vipul: 3 revisions

2008-05-12T16:23:18Z

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Vipul at 21:49, 8 February 2008

2008-02-08T21:49:51Z

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			{{analogue in-of\|graded ring\|ring subset\|ideal}}

	==Definition==		==Definition==

Vipul at 21:44, 8 February 2008

2008-02-08T21:44:28Z

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	===Symbol-free definition===		===Symbol-free definition===

	An [[ideal]] in a [[graded ring]] is termed a '''homogeneous ideal''' if it satisfies the following equivalent conditions:		An [[ideal]] in a [[graded ring]] is termed a '''homogeneous ideal''' or a '''graded ideal''' if it satisfies the following equivalent conditions:

			* It is generated by [[homogeneous element]]s
			* It equals the sum of its intersections with all the homogeneous components (or graded components)
			* It is a graded submodule of the graded ring, viewed as a [[graded module]] over itself

			===Definition with symbols===

			Let <math>A</math> be a [[graded ring]] where the <math>d^{th}</math> graded component is denoted <math>A_d</math>. Then, an ideal <math>I \le A</math> is termed a '''homogeneous ideal''' or '''graded ideal''' if it satisfies the following conditions:

			* <math>I</math> is generated (as an <math>A</math>-module) by homogeneous elements
			* <math>I = \bigoplus_{d=0}^\infty I \cap A_d</math>. The intersection <math>I \cap A_d</math> is denoted <math>I_d</math>, and is the <math>d^{th}</math> graded component of <matH>I</math>, viewed as a graded <math>A</math>-module.

Vipul: New page: ==Definition== ===Symbol-free definition=== An ideal in a graded ring is termed a '''homogeneous ideal''' if it satisfies the following equivalent conditions:

2008-02-08T21:30:34Z

New page: ==Definition== ===Symbol-free definition=== An ideal in a graded ring is termed a '''homogeneous ideal''' if it satisfies the following equivalent conditions:

New page

==Definition==

===Symbol-free definition===

An [[ideal]] in a [[graded ring]] is termed a '''homogeneous ideal''' if it satisfies the following equivalent conditions: