Localization of injective modules over Pr\"ufer domains

It is proved that S −1 G is injective if G is an injective module of finite Goldie dimension over a reduced arithmetic ring R, for each multi-plicative subset S. Moreover, if R is a Prüfer domain of finite character then localizations of injective modules are injective too. If R is a noetherian or hereditary ring, it is well known that localizations of injective R-modules are injective. By [1, Corollary 8] this property holds if R is a h-local Prüfer domain. However [1, Example 1] shows that this result is not generally true. Moreover, by [2, Theorem 25] there exist a coherent domain R, a multiplicative subset S and an injective module G such that S −1 G is not injective. The aim of this paper is to study localizations of injective modules over arithmetic rings. We deduce from [1, Theorem 3] the two following results: any local-ization of an injective module of finite Goldie dimension over a reduced arithmetic ring is injective (Theorem 1) and any localization of an injective module over a Prüfer domain of finite character is injective (Theorem 3). 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 arithmetic if R P is a valuation ring for every maximal ideal P. An arithmetic 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.