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-module is a direct sum of cyclic R-modules. In [14], it was also proved that if a Bézout domain satisfies Property (N), then it is an elementary divisor domain. The aim of this article is to generalize this result (as well as others) to a much wider class of rings. Our main result is that a Bézout ring all whose proper homomorphic images have stable range 1 (in particular, a neat ring) is an elementary divisor ring.