# Reduction matrices over Bezout rings

- Written by Andrij Sagan
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*Department of Algebra and Logic** Faculty of Mechanics and Mathematics** Ivan Franko National University of L'vivemail: andrij.sagan at gmail.com*

**Abstract:**

*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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