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Torsion of injective modules and weakly pro-regular sequences
- Source :
- Communications in Algebra. 48:3637-3650
- Publication Year :
- 2020
- Publisher :
- Informa UK Limited, 2020.
-
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.<br />Comment: 14 pages, to appear in Comm. in Algebra
- Subjects :
- Noetherian
Pure mathematics
Algebra and Number Theory
Mathematics::Commutative Algebra
010102 general mathematics
010103 numerical & computational mathematics
Commutative ring
Commutative Algebra (math.AC)
Mathematics - Commutative Algebra
01 natural sciences
Injective module
Primary 13C11, Secondary 13B80, 13E05
Injective function
Multiplicatively closed set
FOS: Mathematics
Torsion (algebra)
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 15324125 and 00927872
- Volume :
- 48
- Database :
- OpenAIRE
- Journal :
- Communications in Algebra
- Accession number :
- edsair.doi.dedup.....9e4ef35e4008c2d0250cd660f60916fe