Andriy SaganDepartment of Algebra and Logic 

May 16, 2017 at 15:05 in Lecture Room 377 
Abstract of talk
It this talk the author will present main results of his PhD thesis.
We will discuss an elementary matrix reduction over different classes of commutative and noncommutative rings. In particular, we will specify the necessary and sufficient conditions for a quasiEuclidean duoring being a ring with elementary matrix reduction. Using this criterion we will be able to describe various duorings with elementary matrix reduction. Moreover, it will be established that any right Hermite stable range one ring is a right Euclidean domain.
Additionally, it will be 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 will introduce the concept of eatomic commutative ring and describe its main properties. Furthermore, we will show that any eatomic Bezout domain and locally eatomic Bezout ring are rings with elementary matrix reduction.
Finally, we will prove that a integral domain is an Euclidean ring if and only if a ring of formal Laurent series is an Euclidean ring.
Andriy Sagan
May 16, 2017