Wednesday 16 November 2016 h. 14:30, Room 2BC30

Laura Cossu (Padova, Dip. Mat.)

“Products of elementary and idempotent matrices and non-euclidean pids”

Abstract

It is well known that Gauss Elimination produces a factorization into elementary matrices of any invertible matrix over a field. Is it possible to characterize integral domains different from fields that satisfy the same property? As a partial answer, in 1993, Ruitenburg proved that in the class of Bézout domains, any invertible matrix can be written as a product of elementary matrices if and only if any singular matrix can be written as a product of idempotents.

In this seminar we present some classical results on these factorization properties and we focus, in particular, on their connection with the notion of weak-Euclidean algorithm. We then conclude with a conjecture on non-Euclidean principal ideal domains, rare and interesting objects in commutative algebra, and some related results.

In order to make the talk understandable to a general audience, we will recall basic definitions of Commutative Ring theory and provide easy examples of the objects involved.