#### Diagonal reduction of matrices over Bezout rings with stable 1

February 22, 2019 at 15:05 in Lecture Room 372

We will consider the diagonal reduction of matrices over noncommutative Bezout rings of stable range 1 with condition Kazimirsky.

February 22, 2019 at 15:05 in Lecture Room 372

We will consider the diagonal reduction of matrices over noncommutative Bezout rings of stable range 1 with condition Kazimirsky.

#### Elementary reduction of matrices over Bezout duo-rings

Theorem 1. Every right semihereditary Bezout duo-ring with stable range 1 is a ring with elementary reduction of matrices. Theorem 2. Let R be a right semihereditary duo-ring. Then the following are equivalent: 1) R is a quasi-euclidean ring; 2) R is a Bezout ring and GLn(R) = GEn(R) for every positive integer n≥2.

#### Commutative Bezout rings in which 0 is adequate is a semi regular

Theorem 1. A ring R is a commutative ring in which 0 is adequate. Then for 0∈R/aR an element 0 is adequate. Theorem 2. A ring R is a commutative Bezout ring in which 0 is adequate. Then for any nonzero and any noninvertible alement b∈R there exist an idempotent e∈R, such that be∈J(R) and eR+bR=R. Theorem 3. Semi prime commutative Bezout ring R is a ring in which zero is adequate if and only if R is a regular ring. Theorem…

#### Common divisors and common multiples of singular matrices over elementary diviso…

Ph.D. in materials

#### A criterion of elementary divisor domain for distributive domains

Theorem 1. Any distributive Bezout domain is an elementary divisor domain if and only if it is a duo-domain of neat range 1.

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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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E-mail: oromaniv at franko.lviv.ua