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PreprintCC BY-NC-ND 4.0arXiv

On the Quotient of a Pseudo-null Module

  • Peikai Qi
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Abstract

Motivated by questions arising in noncommutative Iwasawa theory, let R be a (not necessarily commutative) ring, T ∈ R a regular central element, and M a pseudo-null R-module. We investigate necessary and sufficient conditions under which the quotient M/TM is pseudo-null as an R/TR-module. We first give necessary and sufficient conditions in terms of associated prime ideals when R is commutative. Then we apply this to the case when R is a Krull domain and obtain a precise relationship between the characteristic ideals of M/TM and the T-torsion submodule M[T]. We give necessary and sufficient conditions in terms of Ext groups when R is a noncommutative ring and then compare them with the criterion when R is commutative. Lastly, we give an application to noncommutative Iwasawa theory. Roughly speaking, if `big' dual fine Selmer group is pseudo-null, then `most' specialized dual fine Selmer group is pseudo-null.

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Cite

BibTeXRIS
BibTeX
@misc{qi2026quotient,
  title = {{On the Quotient of a Pseudo-null Module}},
  author = {Peikai Qi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39655},
  eprint = {2609.39655},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39655},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On the Quotient of a Pseudo-null Module
AU  - Peikai Qi
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39655
UR  - https://arxiv.org/abs/2609.39655
AB  - Motivated by questions arising in noncommutative Iwasawa theory, let R be a (not necessarily commutative) ring, T ∈ R a regular central element, and M a pseudo-null R-module. We investigate necessary and sufficient conditions under which the quotient M/TM is pseudo-null as an R/TR-module. We first give necessary and sufficient conditions in terms of associated prime ideals when R is commutative. Then we apply this to the case when R is a Krull domain and obtain a precise relationship between the characteristic ideals of M/TM and the T-torsion submodule M[T]. We give necessary and sufficient conditions in terms of Ext groups when R is a noncommutative ring and then compare them with the criterion when R is commutative. Lastly, we give an application to noncommutative Iwasawa theory. Roughly speaking, if `big' dual fine Selmer group is pseudo-null, then `most' specialized dual fine Selmer group is pseudo-null.
ER  -