Injective modules and semistar operations

We approach the problem of classifying injective modules over an integral domain, by considering the class of semistar Noetherian domains. When working with such domains, one has to focus on semistar ideals: as a consequence for modules, we restrict our study to the class of injective hulls of co-semistar modules, those in which the annihilator ideal of each nonzero element is semistar. We obtain a complete classification of this class, by describing its elements as injective hulls of uniquely determined direct sums of indecomposable injective modules; if moreover, we consider stable semistar operations, then we can further improve this result, obtaining a natural generalization of the classical Noetherian case. Our approach provides a unified treatment of results on injective modules over various kinds of domains obtained by Matlis, Cailleau, Beck, Fuchs and Kim–Kim–Park.