Elementary divisor ring

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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

A commutative unital ring $R$ is termed an elementary divisor ring if for every matrix $M$ (not necessarily square) with entries in $R$, there exist invertible square matrices $P$ and $Q$ such that $PMQ$ is a diagonal matrix where the $i^{th}$ diagonal entry divides the $(i+1{)}^{th}$ diagonal entry.

Relation with other properties

Stronger properties

• Principal ideal ring
• Elementary divisor domain

Weaker properties

• Hermite ring