Bezout duo ring R is an elementary divisor ring iff R is a ring of neat range 1

Abstract

Abstract. All rings are associative rings with nonzero identity. If every matrix over R admits a canonical diagonal reduction then R is said to be an elementary divisor ring. Right (left) Bezout rings are rings whose finitely generated right ideals are principal right (left) ideals. Bezout ring is a ring which is a right and left Bezout ring. A ring R is said to be a duo ring if every right or left one-sided ideal in R is two-sided. A ring R is said to have stable range 1, if for any a, b ∈ R such that aR + bR = R there exists t ∈ R such that (a + bt)R = R. A ring R is said to have stable range 2 if for all a, b, c ∈ R such that aR+bR+cR = R, there exists x, y ∈ R such that (a+cx)R+(b+cy)R = R. A ring R is said to be a ring of neat range 1 if for any elements a, b ∈ R such that aR+bR = R and for any nonzero element c ∈ R there exist such elements u, v, t ∈ R that a+bt = uv, where uR+cR = R, vR+(1−c)R = R, and uR + vR = R.