A commutative ring with identity is a valuation ring if its lattice of ideals is totally ordered by inclusion, it is a valuation domain if in addition it has no zero divisors. Clearly epic images of valuation domains are valuation rings. Some time ago, I. Kaplansky raised the question if all valuation rings are of this form. Only quite recently a partial answer could be given by L. Fuchs and L. Salce in their book “Modules over valuation domains” (1985; Zbl 0578.13004): In the constructible universe they exhibited a valuation ring of cardinality \(\omega_ 1\), which is not an epic image of a valuation domain. This was carried out by applying Jensen’s combinatorial principle \(\diamond_{\omega_ 1}.\)
The purpose of this note is to show that a set theoretic assumption weaker than the continuum hypothesis suffices, namely \(2^{\aleph_ 0}<2^{\aleph_ 1}\).