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Reduced and coreduced modules with respect to inverse families of ideals

  • Sholastica Luambano
  • David Ssevviiri
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Abstract

Let R be a commutative ring and Φ an inverse family of ideals with greatest proper member ℒ. We introduce and study Φ-reduced and Φ-coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized torsion and completion functors associated with Φ are determined by ℒ. We establish characterizations and closure properties of these modules and obtain a Greenlees-May type adjunction and a Matlis-Greenlees-May type characterization. When Φ is a system of ideals, we further investigate the radicality of the generalized torsion functor and derive associated torsion theories on suitable Serre subcategories. These results provide a module-theoretic framework for studying generalized torsion and completion with respect to families of ideals.

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Cite

BibTeXRIS
BibTeX
@misc{luambano2026reduced,
  title = {{Reduced and coreduced modules with respect to inverse families of ideals}},
  author = {Sholastica Luambano and David Ssevviiri},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30091},
  eprint = {2609.30091},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30091},
}
RIS
TY  - UNPB
TI  - Reduced and coreduced modules with respect to inverse families of ideals
AU  - Sholastica Luambano
AU  - David Ssevviiri
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30091
UR  - https://arxiv.org/abs/2609.30091
AB  - Let R be a commutative ring and Φ an inverse family of ideals with greatest proper member ℒ. We introduce and study Φ-reduced and Φ-coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized torsion and completion functors associated with Φ are determined by ℒ. We establish characterizations and closure properties of these modules and obtain a Greenlees-May type adjunction and a Matlis-Greenlees-May type characterization. When Φ is a system of ideals, we further investigate the radicality of the generalized torsion functor and derive associated torsion theories on suitable Serre subcategories. These results provide a module-theoretic framework for studying generalized torsion and completion with respect to families of ideals.
ER  -