Torsion of injective modules and weakly pro-regular sequences. Issue 9 (1st September 2020) - Schenzel, Peter; Simon, Anne-Marie
Abstract: Let R a commutative ring, a ⊂ R an ideal, I an injective R -module and S ⊂ R a multiplicatively closed set. When R is Noetherian it is well-known that the a -torsion sub-module Γ a ( I ), the factor module I / Γ a ( I ) and the localization IS are again injective R -modules. We investigate these properties in the case of a commutative ring R by means of a notion of relatively- 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.