Torsion of injective modules and weakly pro-regular sequences

Let $R$ a commutative ring, $\mathfrak{a}\subset R$ an ideal, $I$ an injective $R$-module and $S\subset R$ a multiplicatively closed set. When $R$ is Noetherian it is well-known that the $\mathfrak{a}$-torsion sub-module ${\mathrm{\Gamma }}_{\mathfrak{a}}(I)$, the factor module $I\mathrm{/}{\mathrm{\Gamma }}_{\mathfrak{a}}(I)$ 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-$\mathfrak{a}$-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.