Torsion of injective modules and weakly pro-regular sequences
2020
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 $\Gamma_{\mathfrak{a}}(I)$,
the factor module $I/\Gamma_{\mathfrak{a}}(I)$ 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.
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