Torsion of injective modules and weakly pro-regular sequences

Abstract

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.

Keywords:

• Injective module
• non-Noetherian commutative ring
• torsion
• weakly pro-regular sequences

2010 Mathematics Subject Classification:

• Primary: 13C11
• Secondary: 13B30
• 13E05