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A note on inverse limits of injective modules

  • Chencheng Zhang
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Abstract

Let R be a ring and M a left R-module. We prove that M is an ω1-indexed inverse limit of injective modules with split epimorphic connecting maps, and M is also an ω-indexed inverse limit of a stationary tower defined by a split monic endomorphism of one injective module. These constructions answer two questions of Bergman. We then study the class of direct summands of ω-indexed inverse limit of injective modules with epimorphic connecting maps. Over a commutative Noetherian ring R, they are precisely the Enochs cotorsion modules if and only if gldimR≤1 and SpecR is countable.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026note,
  title = {{A note on inverse limits of injective modules}},
  author = {Chencheng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36089},
  eprint = {2609.36089},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36089},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - A note on inverse limits of injective modules
AU  - Chencheng Zhang
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36089
UR  - https://arxiv.org/abs/2609.36089
AB  - Let R be a ring and M a left R-module. We prove that M is an ω_1-indexed inverse limit of injective modules with split epimorphic connecting maps, and M is also an ω-indexed inverse limit of a stationary tower defined by a split monic endomorphism of one injective module. These constructions answer two questions of Bergman. We then study the class of direct summands of ω-indexed inverse limit of injective modules with epimorphic connecting maps. Over a commutative Noetherian ring R, they are precisely the Enochs cotorsion modules if and only if gldimR≤1 and SpecR is countable.
ER  -