Sagan A. Reduction of matrices over Bezout rings.

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