"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

Reduction matrices over Bezout rings

In particular, we will specify the necessary and sufficient conditions for a quasi-Euclidean duo-ring being a ring with elementary matrix reduction. Using this criterion we will be able to describe various duo-rings with elementary matrix reduction. Moreover, it will be established that any right Hermite stable range one ring is a right ω-Euclidean domain. Additionally, it will be proved that for any pair of nxn full matrices over elementary divisor ring, there exists a right (left) divisibility chain of length 2(n-1) and over PID – of length 2. Among the other results we will introduce the concept of e-atomic commutative domain and describe its main properties. Furthermore, we will show that any e-atomic Bezout domain and locally e-atomic Bezout domain are rings with elementary matrix reduction. Among the other results we will introduce the concept of EID-ring and describe its main properties. Shown that commutative ring with elementary reduction of matrices is EID-ring. Proved that commutative ω-Euclidean domain is a ring of elementary reduction matrices if and only if for each ideal I the ring R/I is EID-ring. Finally, we will prove that a integral domain R is an ω-Euclidean ring if and only if a ring of formal Laurent series  is an ω-Euclidean ring. It is shown that an arbitrary degenerate matrix over the ring of formal power series Laurent, where the ring is ω-Euclidean domain coefficients, is recorded as the product idempotent matrices.

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Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
1 Universytetska Str., 79000 Lviv, Ukraine
Tel: (+380 322) 394 172
E-mail: oromaniv at

Scientific seminar, August, 2017