Valuation domains whose products of free modules are separable

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separable if and only if R is maximal or discrete of rank one. In [3, Theorem XVI.5.4], Fuchs and Salce gave a slight generalization of this result and showed that R is separable if and only if R is discrete of rank one, when R is slender. The aim of this paper is to give another generalization of Franzen’s result by proving Theorem 8 below. If the maximal ideal P is principal, we get that R can be separable when R is neither maximal nor discrete of rank one. This is a negative answer to [3, Problem 59]. For proving his result, Franzen began by showing that each archimedean valuation domain which is not almost maximal, possesses an indecomposable reflexive module of rank 2. We use a similar argument in the proof of Theorem 8. Finally we give an example of a non-archimedean nonslender valuation domain such that R is not separable. This is a positive answer to [3, Problem 58]. In the sequel, R is a commutative unitary ring. An R-module whose submodules are totally ordered by inclusion, is said to be uniserial. IfR is a uniserialR-module, we say that R is a valuation ring. The R-topology of R is the linear topology for which each non-zero ideal is a neighborhood of 0. When R is a valuation ring with maximal ideal P and A is a proper ideal, then R/A is Hausdorff in the R/A-topology if and only if A 6= Pa, ∀0 6= a ∈ R. We say that R is (almost) maximal if R/A is complete in the R/A-topology for each (non-zero) proper ideal A 6= Pa, ∀0 6= a ∈ R. From now on, R is a valuation domain, P is its maximal ideal and Q is its field of quotients. Let M be an R-module and let N be a submodule. We say that N is a pure submodule of M if rN = rM ∩ N, ∀r ∈ R. Let M be a torsion-free module. We say that M is separable if each pure uniserial submodule is a summand. Recall that each element x of M is contained in a pure uniserial submodule U , where U is the inverse image of the torsion submodule of M/Rx by the canonical map M → M/Rx. Let M be a non-zero R-module. As in [3] we set: M ♯ = {s ∈ R | sM ⊂ M}. 2000 Mathematics Subject Classification. 13C10, 13F99, 13G05.