Product-trace-rings and a question of G. S. Garfinkel

It is an open question as to whether every left coherent ring R satisfying the intersection property for finitely generated left ideals of R is a right-product-trace-ring or not. R is a right-product-trace-ring iff every product of trace-right-R-modules (= universally torsionless-right-R-modules) is a trace-right-R-module. This question is shown to have a negative answer. Furthermore, looking at all valuation domains, the complete product-tracerings, the product-trace-rings and the product-content-rings are characterized.