# Romaniv A. M.

### Greatest common divisor of matrices one of which is a disappear matrix

Romaniv A. M., Shchedrik V. P.

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 425–430

We study the structure of the greatest common divisor of matrices one of which is a disappear matrix. In this connection, we indicate the Smith normal form and the transforming matrices of the left greatest common divisor.

### Elementary Reduction of Matrices over Right 2-Euclidean Rings

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1717 – 1721

We introduce a concept of noncommutative (right) 2-Euclidean ring. We prove that a 2-Euclidean ring is a right Hermite ring, a right Bezout ring, and a *GE* _{ n }-ring. It is shown that an arbitrary right unimodular string of length not less than 3 over a right Bezout ring of stable rank possesses an elementary diagonal reduction. We prove that a right Bezout ring of stable rank 1 is a right 2-Euclidean ring.

### Rings with Elementary Reduction of Matrices

Romaniv A. M., Zabavskii B. V.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1641-1649

We establish necessary and sufficient conditions under which a quasi-Euclidean ring coincides with a ring with elementary reduction of matrices. We prove that a semilocal Bézout ring is a ring with elementary reduction of matrices and show that a 2-stage Euclidean domain is also a ring with elementary reduction of matrices. We formulate and prove a criterion for the existence of solutions of a matrix equation of a special type and write these solutions in an explicit form.