Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
Under a ring we understand an associative ring with 1.
The matrices A and B over a ring R are equivalent if exist invertible matrices P and Q of appropriate sizes such that B=PAQ.
A ring R is called clean if every element of R is the sum of an idempotent and a unit.
A ring R is called an exchange ring if for every element a∈R there exists an idempotent e∈R such that e∈aR, 1-e∈(1-aR).
A ring R is called right (left) distributive if every lattice right (left) ideal of ring R is distributive.
We say that a duo-ring R has neat range 1 if for every a,b∈R such that aR+bR=R there exist an element t∈R such that a factor-ring R/(a+br)R is a clean ring.
Theorem 1. Any distributive Bezout domain is an elementary divisor domain if and only if it is a duo-domain of neat range 1.
 Zabavsky B.V. Diagonal reduction of matrices over rings. Math. Stud., Monograph Series, v. XVI, Lviv, 2012, 251 p.