Torsion of injective modules and weakly pro-regular sequences

Let R a commutative ring, an ideal, I an injective R-module and a multiplicatively closed set. When R is Noetherian it is well-known that the -torsion sub-module the factor module and the localization I-S are again injective R-modules. We investigate these properties in the case of a commutative ring R by means of a notion of relatively--injective R-modules. In particular we get another characterization of weakly pro-regular sequences in terms of relatively injective modules. Also we present examples of non-Noetherian commutative rings R and injective R-modules for which the previous properties do not hold. Moreover, under some weak pro-regularity conditions we obtain results of Mayer-Vietoris type.