On Bézout Domains

Recall that a commutative domain R in which every finitely generated ideal is principal is called a Bézout domain. By definition, a noetherian Bézout domain is a principal ideal domain. Several examples of non-noetherian Bézout domains are listed in [1], 243-246. Recall also that a commutative domain R is called an Elementary Divisor domain if, given any matrix A with coefficients in R, there exist invertible matrices P,Q with coefficients in R such that PAQ = D with D = diag(d1, . . . ) a diagonal matrix (such a matrix may be rectangular, but it has d1, . . . on the main diagonal and zeroes elsewhere). Kaplansky showed in [7], 5.2, that a Bézout domain is an Elementary Divisor domain if and only if it satisfies (∗): For all a, b, c ∈ R with (a, b, c) = R, there exist p, q ∈ R such that (pa, pb + qc) = R (see also [4], 6.3). It is well-known that a principal ideal domain is an Elementary Divisor domain. Consideration of the Elementary Divisor problem for a non-noetherian ring can be found as early as [11]. It is an open question dating back to Helmer [5] in 1942 to decide whether a Bézout domain is always an Elementary Divisor domain. Leavitt and Mosbo in fact state in [8], Remark 8, that it has been conjectured that there exists a Bézout domain that is not an Elementary Divisor domain (see also Problem 5 in [4], p. 122). Our contribution to this question is the introduction of new chains of implications between R is an Elementary Divisor domain and R is Bézout, which may prove useful in an eventual solution to the above open question. Let Mn(R) denote the ring of (n × n)-matrices with coefficients in R. We make the following definitions. Definition 1 Let n ≥ 1. A ring R is called an (SU)n-ring if, given any A ∈ Mn(R), there exist a symmetric matrix S ∈ Mn(R) and an invertible matrix U ∈ GLn(R) such that A = SU . If R is an (SU)n-ring for all n ≥ 1, we shall say that R is an SU-ring. A ring R is called an (SU )n-ring if, given any A ∈ Mn(R), there exist a symmetric matrix S ∈ Mn(R) and an invertible matrix U ∈ SLn(R) such that A = SU . If R is an (SU )n-ring for all n ≥ 1, we shall say that R is an SU ′-ring.