Elementary divisor ring

Origin
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.

Stronger properties

 * Principal ideal ring
 * Elementary divisor domain

Weaker properties

 * Hermite ring