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, given any commutative ring R, new Bézout rings associated with R.