Elementary divisor domain: Difference between revisions

Revision as of 21:26, 5 January 2008

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

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

Fill this in later

Relation with other properties

Stronger properties

Principal ideal domain

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 ==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==