## Factorization into radical ideals Bezout domains

- Written by Bohdan Zabavsky

- Written by Andrij Sagan

*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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V. Shchedryk, 6 May, 2016

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 franko.lviv.ua