Products of elementary matrices and non-Euclidean principal ideal domains

A classical problem, originated by Cohn's 1966 paper [1], is to characterize the integral domains R satisfying the property: (GEn) “every invertible n×n matrix with entries in R is a product of elementary matrices”. Cohn called these rings generalized Euclidean, since the classical Euclidean rings do satisfy (GEn) for every n>0. Important results on algebraic number fields motivated a natural conjecture: a non-Euclidean principal ideal domain R does not satisfy (GEn) for some n>0. We verify this conjecture for two important classes of non-Euclidean principal ideal domains: (1) the coordinate rings of special algebraic curves, among them the elliptic curves having only one rational point; (2) the non-Euclidean PID's constructed by a fixed procedure, described in Anderson's 1988 paper [2].