# A criterion of elementary divisor domain for distributive domains

- Written by Vasylyna Bokhonko
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Department of Algebra and Logic* Faculty of Mechanics and Mathematics** Ivan Franko National University of L'viv*

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Abstract:

*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.*

**References**

[1] Zabavsky B.V. Diagonal reduction of matrices over rings. Math. Stud., Monograph Series, v. XVI, Lviv, 2012, 251 p.