when "a" is a euclidean domain (e.g. a field, or the integers) sk("a") vanishes, and the determinant map is an isomorphism from k("a") to "a". this is "false" in general for pids, thus providing one of the rare mathematical features of euclidean domains that do not generalize to all pids. an explicit pid such that sk is nonzero was given by ischebeck in 1980 and by grayson in 1981. if "a" is a dedekind domain whose quotient field is an algebraic number field (a finite extension of the rationals) then shows that sk("a") vanishes.