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

History

Origin

This term was introduced by: Gillman

This term was introduced by: Henriksen

The original notion of elementary divisor domain was introduced by Kaplansky, and the generalization from integral domains to arbitrary commutative unital rings was done by Gillman and Henriksen in their paper Some remarks about elementary divisor rings.

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

References

• Some remarks about elementary divisor rings by L. Gillman and M. Henriksen, Trans. Amer. Math. Society, 82 (1956), 362-365