Localization of injective modules over arithmetical rings

Francois Couchot (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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 $P$, $R_P$ 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.

Mots clés: arithmetical ring; semicoherent ring; injective module; FP-injective module; finitely injective module; Goldie dimension; valuation ring; Prüfer domain; finite character

Domaines: Anneaux et algèbres [math.RA]

Francois Couchot. Localization of injective modules over arithmetical rings. Communications in Algebra, 2009, 37 (10), pp.3418-3423. ⟨hal-00142184v2⟩