Elementary divisor domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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Definition

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

Relation with other properties

Stronger properties

• Principal ideal domain