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. In the first part of this paper we study the valuation rings that satisfy the following condition (C): every fp-injective module is locally injective. In his paper [5], Alberto Facchini constructs an example of an almost maximal valuation domain satisfying (C) which is not noetherian and gives a negative answer to the following question asked in [1] by Goro Azumaya: if R is a ring that satisfies (C), is R a left noetherian ring? From [5, Theorem 5] we easily deduce that a valuation domain R satisfies (C) if and only if R is almost maximal and its classical field of fractions is countably generated. In this case every indecomposable injective R-module is countably generated. So, when an almost maximal valuation ring R, with eventually non-zero zerodivisors, verifies this last condition, we prove that R satisfies (C). Conversely, every valuation ring that satisfies (C) is almost maximal. In the second part of this paper, we prove that every locally injective module is a factor module of a direct sum of indecomposable injective modules modulo a pure submodule. This result allows us to give equivalent conditions for a valuation ring R to be an IF-ring, i.e. a ring for which every injective R-module is flat. It is proved that each proper localization of Q, the classical ring of fractions of R, is an IF-ring. It is well known that a valuation domain R is almost maximal if and only if the injective dimension of the R-module R is less or equal to one. This result is due to E. Matlis. See [12, Theorem 4]. In the third part, some generalizations of this result are given. Moreover, when the subset Z of zerodivisors of an almost maximal valuation ring R is nilpotent, we show that every uniserial R-module is “standard”(see [7, p.141]). In the last part of this paper we determine some sufficient and necessary conditions for every indecomposable injective module over a valuation ring R to be countably generated. In particular the following condition is sufficient: Spec(R) is a countable set. Moreover, when this condition is satisfied, we prove that every ideal of R is countably generated and that every finitely generated R-module is countably cogenerated.