Gelfand local Bezout domains are elementary divisor rings

Elementary Divisor Domains and Bézout Domains

It is well-known that an Elementary Divisor domain R is a Bézout domain, and it is a classical open question to determine whether the converse statement is false. In this article, we provide new chains of implications between R is an Elementary Divisor domain and R is Bézout defined by hyperplane conditions in the general linear group. Motivated by these new chains of implications, we construct...

Some Remarks About Elementary Divisor Rings

Some Remarks on Elementary Divisor Rings II

Bézout Rings with Almost Stable Range 1 are Elementary Divisor Rings

Abstract. In this article we revisit a problem regarding Bézout domains, namely, whether every Bézout domain is an elementary divisor domain. Elementary divisor domains where defined by Kaplansky [13] and generalized to rings with zero-divisors by Gillman and Henriksen [7]. Later, in [14] it was shown that a domain R is an elementary divisor domain if and only if every finitely presented R-modu...

Formalized linear algebra over Elementary Divisor Rings in Coq

This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to...