Localization of injective modules over arithmetical rings

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P , RP is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreover, if R is a Prüfer domain of finite character, localizations of injective R-modules are injective. This is a sequel and a complement of (Couchot, 2006). If R is a noetherian or hereditary ring, it is well known that localizations of injective R-modules are injective. By (Couchot, 2006, Corollary 8) this property holds if R is a h-local Prüfer domain. However (Couchot, 2006, Example 1) shows that this result is not generally true. E. C. Dade was probably the first to study localizations of injective modules. By (Dade, 1981, Theorem 25), there exist a ring R, a multiplicative subset S and an injective module G such that SG is not injective. In this example we can choose R to be a coherent domain. The aim of this paper is to study localizations of injective modules over arithmetical rings. We deduce from (Couchot, 2006, Theorem 3) the two following results: any localization of an injective R-module of finite Goldie dimension is injective if and only if any localization at a maximal ideal of R is either coherent or non-semicoherent (Theorem 5) and each localization of any injective module over a Prüfer domain of finite character is injective (Theorem 10). Moreover, if any localization at a maximal ideal of R is either coherent or non-semicoherent, and if each finitely generated R-module has a finite Goldie dimension, then each localization of any finitely injective R-module is finitely injective. In this paper all rings are associative and commutative with unity and all modules are unital. A module is said to be uniserial if its submodules are linearly ordered by inclusion. A ring R is a valuation ring if it is uniserial as R-module and R is arithmetical if RP is a valuation ring for every maximal ideal P. An arithmetical domain R is said to be Prüfer. We say that a module M is of Goldie dimension n if and only if its injective hull E(M) is a direct sum of n indecomposable injective modules. We say that a domain R is of finite character if every non-zero element is contained in finitely many maximal ideals. As in (Ramamurthi and Rangaswamy, 1973), a module M over a ring R is said to be finitely injective if every homomorphism f : A → M extends to B whenever A is a finitely generated submodule of an arbitrary R-module B. 2000 Mathematics Subject Classification. Primary 13F05, 13C11.