Matrix reduction over Bezout rings


Andriy Sagan

Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

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 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 \omega-Euclidean domain.

Additionally, it will be proved that for any pair of n\times n 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 ring and describe its main properties. Furthermore, we will show that any e-atomic Bezout domain and locally e-atomic Bezout ring are rings with elementary matrix reduction.

Finally, we will prove that a integral domain R is an \omega-Euclidean ring if and only if a ring of formal Laurent series R [[ x,x^{-1}]] is an \omega-Euclidean ring.

Andriy Sagan
May 16, 2017