Injective modules and fp-injective modules over valuation rings

It is shown that each almost maximal valuation ring R, such that every indecomposable injective R-module is countably generated, satisfies the following condition (C): each fp-injective R-module is locally injective. The converse holds if R is a domain. Moreover, it is proved that a valuation ring R that satisfies this condition (C) is almost maximal. The converse holds if Spec(R) is countable. When this last condition is satisfied it is also proved that every ideal of R is countably generated. New criteria for a valuation ring to be almost maximal are given. They generalize the criterion given by E. Matlis in the domain case. Necessary and sufficient conditions for a valuation ring to be an IF-ring are also given. (C) 2003 Elsevier Inc. All rights reserved.