Elementary divisor domain - Revision history

Vipul: 3 revisions

2008-05-12T16:19:50Z

3 revisions

← Older revision	Revision as of 16:19, 12 May 2008
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Vipul at 21:44, 5 January 2008

2008-01-05T21:44:44Z

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 {{integral domain property}}
 ==Definition==
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 * [[Principal ideal domain]]

Vipul at 21:26, 5 January 2008

2008-01-05T21:26:42Z

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	==Definition==		==Definition==

	An ~~[[integral domain]] <math>R</math> is termed an~~ '''elementary divisor domain''' ~~if, given~~ any ~~positive integer <math>n</math> and any matrix <math>A</math>~~ of order <math>n</math> over <math>R</math>, there exist invertible matrices <math>U</math> and <math>V</math> of order <math>n</math>, such that <math>UAV</math> is a diagonal matrix with diagonal entries <math>s_1(A), s_2(A), \ldots, s_n(A)</math>, such that <math>s_i(A)\|s_{i+1}(A)</math>. In other words, <math>R</math> is ~~termed~~ an ~~'''~~elementary divisor domain~~''' if every matrix admits~~ a [[~~Smith normal form~~]].		===Symbol-free definition===

			An '''elementary divisor domain''' or '''Hermite domain''' is any of the following equivalent things:

			* An [[integral domain]] which is also an [[elementary divisor ring]]
			* An [[integral domain]] which is also a [[Hermite ring]]

			===Definition with symbols===

			{{fillin}}

	==Relation with other properties==		==Relation with other properties==

Vipul at 23:34, 16 December 2007

2007-12-16T23:34:13Z

New page

{{integral domain property}}

==Definition==

An [[integral domain]] <math>R</math> is termed an '''elementary divisor domain''' if, given any positive integer <math>n</math> and any matrix <math>A</math> of order <math>n</math> over <math>R</math>, there exist invertible matrices <math>U</math> and <math>V</math> of order <math>n</math>, such that <math>UAV</math> is a diagonal matrix with diagonal entries <math>s_1(A), s_2(A), \ldots, s_n(A)</math>, such that <math>s_i(A)|s_{i+1}(A)</math>. In other words, <math>R</math> is termed an '''elementary divisor domain''' if every matrix admits a [[Smith normal form]].

==Relation with other properties==

===Stronger properties===

* [[Principal ideal domain]]