Vipul: 5 revisions

2008-05-12T16:23:09Z

5 revisions

← Older revision	Revision as of 16:23, 12 May 2008
(No difference)

Vipul: Noetherianness is polynomial-closed moved to Hilbert basis theorem over redirect

2008-03-03T19:33:28Z

Noetherianness is polynomial-closed moved to Hilbert basis theorem over redirect

← Older revision	Revision as of 19:33, 3 March 2008
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Vipul at 19:14, 3 March 2008

2008-03-03T19:14:28Z

← Older revision		Revision as of 19:14, 3 March 2008
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			{{Noetherian ring result}}
	{{curing metaproperty satisfaction}}		{{curing metaproperty satisfaction}}

Vipul at 19:14, 3 March 2008

2008-03-03T19:14:14Z

← Older revision		Revision as of 19:14, 3 March 2008
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	{{curing metaproperty satisfaction}}		{{curing metaproperty satisfaction}}

			==Name==

			This result is termed the '''Hilbert basis theorem'''.

	==Statement==		==Statement==

Vipul at 19:17, 7 January 2008

2008-01-07T19:17:27Z

@@ Line 16: / Line 16: @@
 ==Proof==
-The proof rests on the notion of the [[leading coefficient map]] and the fact that if the image of an ideal under the leading coefficient map is finitely generated, so is the original ideal.
+===Proof idea===
-{{further|[[image under leading coefficient map is finitely generated implies finitely generated]]}}
+The proof rests on the notion of the [[leading coefficient map]] and the fact that if the image of an ideal under the leading coefficient map is Noetherian, then so is the original ideal.

Vipul at 17:23, 7 January 2008

2008-01-07T17:23:00Z

New page

{{curing metaproperty satisfaction}}

==Statement==

===Property-theoretic statement===

The [[property of commutative unital rings]] of being [[Noetherian ring|Noetherian]] is [[polynomial-closed property of commutative unital rings|polynomial-closed]].

===Verbal statement===
The [[polynomial ring]] over a [[Noetherian ring]] is Noetherian.

===Symbolic statement===

Let <math>A</math> be a Noetherian ring. Then the polynomial ring <math>A[x]</math> (where <math>x</math> is an indeterminate) is also a Noetherian ring.

==Proof==

The proof rests on the notion of the [[leading coefficient map]] and the fact that if the image of an ideal under the leading coefficient map is finitely generated, so is the original ideal.

{{further|[[image under leading coefficient map is finitely generated implies finitely generated]]}}

Hilbert basis theorem - Revision history

Vipul: 5 revisions

Vipul: Noetherianness is polynomial-closed moved to Hilbert basis theorem over redirect

Vipul at 19:14, 3 March 2008

Vipul at 19:14, 3 March 2008

Vipul at 19:17, 7 January 2008

Vipul at 17:23, 7 January 2008